Theorem 2eu5 1071

Description: An alternate definition of double existential uniqueness (see 2eu4 1070). A mistake sometimes made in the literature is to use E!xE!y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A.xE*yph as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.")

Assertion

Ref	Expression
2eu5	|- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Distinct variable group(s):   x,y,z,w   ph,z,w

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		eumo 1037	. . . . . 6 |- (E!yE.xph -> E*yE.xph)
2	1	adantl 305	. . . . 5 |- ((E!xE.yph /\ E!yE.xph) -> E*yE.xph)
3		2moex 1060	. . . . 5 |- (E*yE.xph -> A.xE*yph)
4	2, 3	syl 12	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> A.xE*yph)
5	4	pm4.71i 483	. . 3 |- ((E!xE.yph /\ E!yE.xph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
6		2eu1 1067	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
7	6	pm5.32ri 490	. . 3 |- ((E!xE!yph /\ A.xE*yph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
8	5, 7	bitr4 154	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E!xE!yph /\ A.xE*yph))
9		2eu4 1070	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
10	8, 9	bitr3 153	1 |- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007  E*wmo 1008

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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