Metamath Proof Explorer - mmtheorems11

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Statement List for Metamath Proof Explorer - 1001-1100 - Page 11 of 58

Type	Label	Description
Statement
Theorem	ddeeq1 1001	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (y = z -> A.x y = z))
Theorem	ddeeq2 1002	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	ddeel1 1003	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (y e. z -> A.x y e. z))
Theorem	ddeel2 1004	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z e. y -> A.x z e. y))
Theorem	sbal2 1005	Move quantifier in and out of substitution.
|- (-. A.x x = y -> ([z / y]A.xph <-> A.x[z / y]ph))
Theorem	ax15 1006	Axiom ax-15 806 is redundant if we assume ax-17 925. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.
|- (-. A.z z = x -> (-. A.z z = y -> (x e. y -> A.z x e. y)))
Syntax	weu 1007	Extend wff definition to include existential uniqueness ("there exists a unique x such that ph").
wff E!xph
Syntax	wmo 1008	Extend wff definition to include uniqueness ("there exists at most one x such that ph").
wff E*xph
Definition	df-eu 1009	Define existential uniqueness, i.e. "there exists exactly one x such that ph." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 1019, eu2 1023, eu3 1024, and eu5 1035 (which in some cases we show with a hypothesis ph -> A.yph in place of a distinct variable condition on y and ph). Double uniqueness is tricky: E!xE!yph does not mean "exactly one x and one y" (see 2eu4 1070).
|- (E!xph <-> E.yA.x(ph <-> x = y))
Definition	df-mo 1010	Define "there exists at most one x such that ph". Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 1027. For other possible definitions see mo2 1026 and mo4 1029.
|- (E*xph <-> (E.xph -> E!xph))
Theorem	euf 1011	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition.
|- (ph -> A.yph)   =>   |- (E!xph <-> E.yA.x(ph <-> x = y))
Theorem	bieud 1012	Formula-building rule for uniqueness quantifier (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (E!xps <-> E!xch))
Theorem	bieudv 1013	Formula-building rule for uniqueness quantifier (deduction rule).
|- (ph -> (ps <-> ch))   =>   |- (ph -> (E!xps <-> E!xch))
Theorem	bieu 1014	Introduce uniqueness quantifier to both sides of an equivalence.
|- (ph <-> ps)   =>   |- (E!xph <-> E!xps)
Theorem	hbeu1 1015	Bound-variable hypothesis builder for uniqueness.
|- (E!xph -> A.xE!xph)
Theorem	hbeu 1016	Bound-variable hypothesis builder for "at most one". Note that x and y needn't be distinct (this makes the proof more difficult).
|- (ph -> A.xph)   =>   |- (E!yph -> A.xE!yph)
Theorem	sb8eu 1017	Variable substitution in uniqueness quantifier. (This theorem can also be proved without requiring that x and y be distinct, but the proof would be longer.)
|- (ph -> A.yph)   =>   |- (E!xph <-> E!y[y / x]ph)
Theorem	cbveu 1018	Rule used to change bound variables with implicit substitution.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E!xph <-> E!yps)
Theorem	eu1 1019	An alternate way of expressing uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.
|- (ph -> A.yph)   =>   |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
Theorem	mo 1020	Equivalent definitions of "there exists at most one".
|- (ph -> A.yph)   =>   |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
Theorem	euex 1021	Existential uniqueness implies existence.
|- (E!xph -> E.xph)
Theorem	eumo0 1022	Existential uniqueness implies "at most one".
|- (ph -> A.yph)   =>   |- (E!xph -> E.yA.x(ph -> x = y))
Theorem	eu2 1023	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.
|- (ph -> A.yph)   =>   |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
Theorem	eu3 1024	An alternate way of expressing existential uniqueness.
|- (ph -> A.yph)   =>   |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
Theorem	euorv 1025	Introduction of a disjunct into uniqueness quantifier.
|- ((-. ph /\ E!xps) -> E!x(ph \/ ps))
Theorem	mo2 1026	Alternate definition of "at most one".
|- (ph -> A.yph)   =>   |- (E*xph <-> E.yA.x(ph -> x = y))
Theorem	mo3 1027	Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis.
|- (ph -> A.yph)   =>   |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
Theorem	mo4f 1028	"At most one" expressed using implicit substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
Theorem	mo4 1029	"At most one" expressed using implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
Theorem	bimod 1030	Formula-building rule for "at most one" quantifier (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (E*xps <-> E*xch))
Theorem	bimo 1031	Formula-building rule for "at most one" quantifier (inference rule).
|- (ps <-> ch)   =>   |- (E*xps <-> E*xch)
Theorem	hbmo1 1032	Bound-variable hypothesis builder for "at most one".
|- (E*xph -> A.xE*xph)
Theorem	hbmo 1033	Bound-variable hypothesis builder for "at most one".
|- (ph -> A.xph)   =>   |- (E*yph -> A.xE*yph)
Theorem	cbvmo 1034	Rule used to change bound variables with implicit substitution.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> E*yps)
Theorem	eu5 1035	Uniqueness in terms of "at most one".
|- (E!xph <-> (E.xph /\ E*xph))
Theorem	eu4 1036	Uniqueness using implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))
Theorem	eumo 1037	Existential uniqueness implies "at most one".
|- (E!xph -> E*xph)
Theorem	eumoi 1038	"At most one" inferred from existential uniqueness.
|- E!xph   =>   |- E*xph
Theorem	exmoeu 1039	Existence in terms of "at most one" and uniqueness.
|- (E.xph <-> (E*xph -> E!xph))
Theorem	exmoeu2 1040	Existence implies "at most one" is equivalent to uniqueness.
|- (E.xph -> (E*xph <-> E!xph))
Theorem	moabs 1041	Absorption of existence condition by "at most one".
|- (E*xph <-> (E.xph -> E*xph))
Theorem	exmo 1042	Something exists or at most one exists.
|- (E.xph \/ E*xph)
Theorem	immo 1043	"At most one" is preserved through implication (notice wff reversal).
|- (A.x(ph -> ps) -> (E*xps -> E*xph))
Theorem	moimv 1044	Move antecedent outside of "at most one".
|- (E*x(ph -> ps) -> (ph -> E*xps))
Theorem	euimmo 1045	Uniqueness implies "at most one" through implication.
|- (A.x(ph -> ps) -> (E!xps -> E*xph))
Theorem	moan 1046	"At most one" is still the case when a conjunct is added.
|- (E*xph -> E*x(ps /\ ph))
Theorem	moani 1047	"At most one" is still true when a conjunct is added.
|- E*xph   =>   |- E*x(ps /\ ph)
Theorem	moor 1048	"At most one" is still the case when a disjunct is removed.
|- (E*x(ph \/ ps) -> E*xph)
Theorem	mooran1 1049	"At most one" imports disjunction to conjunction.
|- ((E*xph \/ E*xps) -> E*x(ph /\ ps))
Theorem	mooran2 1050	"At most one" exports disjunction to conjunction.
|- (E*x(ph \/ ps) -> (E*xph /\ E*xps))
Theorem	moanim 1051	Introduction of a conjunct into "at most one" quantifier.
|- (ph -> A.xph)   =>   |- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	moanimv 1052	Introduction of a conjunct into "at most one" quantifier.
|- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	euanv 1053	Introduction of a conjunct into uniqueness quantifier.
|- (E!x(ph /\ ps) <-> (ph /\ E!xps))
Theorem	mopick 1054	"At most one" picks a variable value, eliminating an existential quantifier.
|- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupick 1055	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps regardless of x. This theorem, which apparently does not appear explicitly in the literature, can be quite useful because it lets us eliminate existential quantifiers in a hypothesis.
|- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupickb 1056	Existential uniqueness "pick" showing wff equivalence.
|- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))
Theorem	mopick2 1057	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 773.
|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))
Theorem	moexex 1058	"At most one" double quantification.
|- (ph -> A.yph)   =>   |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	moexexv 1059	"At most one" double quantification.
|- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	2moex 1060	Double quantification with "at most one".
|- (E*xE.yph -> A.yE*xph)
Theorem	2euex 1061	Double quantification with existential uniqueness.
|- (E!xE.yph -> E.yE!xph)
Theorem	2eumo 1062	Double quantification with existential uniqueness and "at most one".
|- (E!xE*yph -> E*xE!yph)
Theorem	2eu2ex 1063	Double existential uniqueness.
|- (E!xE!yph -> E.xE.yph)
Theorem	2moswap 1064	A condition allowing swap of "at most one" and existential quantifiers.
|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
Theorem	2euswap 1065	A condition allowing swap of uniqueness and existential quantifiers.
|- (A.xE*yph -> (E!xE.yph -> E!yE.xph))
Theorem	2exeu 1066	Double existential uniqueness implies double uniqueness quantification.
|- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
Theorem	2eu1 1067	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
|- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu2 1068	Double existential uniqueness.
|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))
Theorem	2eu3</