Theorem mopick 1054

Description: "At most one" picks a variable value, eliminating an existential quantifier.

Assertion

Ref	Expression
mopick	|- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Proof of Theorem mopick

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- ((ph /\ ps) -> A.y(ph /\ ps))
2		hbs1 986	. . . . . 6 |- ([y / x]ph -> A.x[y / x]ph)
3		hbs1 986	. . . . . 6 |- ([y / x]ps -> A.x[y / x]ps)
4	2, 3	hban 704	. . . . 5 |- (([y / x]ph /\ [y / x]ps) -> A.x([y / x]ph /\ [y / x]ps))
5		sbequ12 865	. . . . . 6 |- (x = y -> (ph <-> [y / x]ph))
6		sbequ12 865	. . . . . 6 |- (x = y -> (ps <-> [y / x]ps))
7	5, 6	anbi12d 476	. . . . 5 |- (x = y -> ((ph /\ ps) <-> ([y / x]ph /\ [y / x]ps)))
8	1, 4, 7	cbvex 849	. . . 4 |- (E.x(ph /\ ps) <-> E.y([y / x]ph /\ [y / x]ps))
9		sbequ2 864	. . . . . . . . . 10 |- (x = y -> ([y / x]ps -> ps))
10	9	syl3 18	. . . . . . . . 9 |- (((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> ([y / x]ps -> ps)))
11	10	exp3a 292	. . . . . . . 8 |- (((ph /\ [y / x]ph) -> x = y) -> (ph -> ([y / x]ph -> ([y / x]ps -> ps))))
12	11	com4t 40	. . . . . . 7 |- ([y / x]ph -> ([y / x]ps -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps))))
13	12	imp 277	. . . . . 6 |- (([y / x]ph /\ [y / x]ps) -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps)))
14		ax-17 925	. . . . . . . 8 |- (ph -> A.yph)
15	14	mo3 1027	. . . . . . 7 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
16		ax-4 673	. . . . . . . 8 |- (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
17	16	a4s 682	. . . . . . 7 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
18	15, 17	sylbi 174	. . . . . 6 |- (E*xph -> ((ph /\ [y / x]ph) -> x = y))
19	13, 18	syl5 22	. . . . 5 |- (([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
20	19	19.23aiv 952	. . . 4 |- (E.y([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
21	8, 20	sylbi 174	. . 3 |- (E.x(ph /\ ps) -> (E*xph -> (ph -> ps)))
22	21	com12 13	. 2 |- (E*xph -> (E.x(ph /\ ps) -> (ph -> ps)))
23	22	imp 277	1 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852  E*wmo 1008

This theorem is referenced by:  eupick 1055  mopick2 1057  moexex 1058  imadif 2714

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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