Theorem 2exeu 2549

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 2499	. . . 4 |- ( E! x E. y ph -> E* x E. y ph )
2		euex 2494	. . . . 5 |- ( E! y ph -> E. y ph )
3	2	moimi 2520	. . . 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 2544	. . 3 |- ( E! y E. x ph -> E. x E! y ph )
6	4, 5	anim12ci 591	. 2 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> ( E. x E! y ph /\ E* x E! y ph ) )
7		eu5 2496	. 2 |- ( E! x E! y ph <-> ( E. x E! y ph /\ E* x E! y ph ) )
8	6, 7	sylibr 224	1 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2eu1  2553  2eu2  2554  2eu3  2555