Theorem 2exeu 2378

Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)

Assertion

Ref	Expression
2exeu	|- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )

Proof of Theorem 2exeu

Step	Hyp	Ref	Expression
1		eumo 2328	. . . 4 |- ( E! x E. y ph -> E* x E. y ph )
2		euex 2323	. . . . 5 |- ( E! y ph -> E. y ph )
3	2	moimi 2349	. . . 4 |- ( E* x E. y ph -> E* x E! y ph )
4	1, 3	syl 17	. . 3 |- ( E! x E. y ph -> E* x E! y ph )
5		2euex 2373	. . 3 |- ( E! y E. x ph -> E. x E! y ph )
6	4, 5	anim12ci 571	. 2 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> ( E. x E! y ph /\ E* x E! y ph ) )
7		eu5 2325	. 2 |- ( E! x E! y ph <-> ( E. x E! y ph /\ E* x E! y ph ) )
8	6, 7	sylibr 216	1 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )

Syntax hints:   -> wi 4   /\ wa 371   E.wex 1663   E!weu 2299   E*wmo 2300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-eu 2303  df-mo 2304

This theorem is referenced by:  2eu1  2382  2eu2  2383  2eu3  2384