Theorem 2euex 2106

Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
2euex	|- ( E! x E. y ph -> E. y E! x ph )

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 2066	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom 1657	. . . 4 |- ( E. x E. y ph <-> E. y E. x ph )
3		hbe1 1488	. . . . . 6 |- ( E. y ph -> A. y E. y ph )
4	3	hbmo 2058	. . . . 5 |- ( E* x E. y ph -> A. y E* x E. y ph )
5		19.8a 1583	. . . . . . 7 |- ( ph -> E. y ph )
6	5	moimi 2084	. . . . . 6 |- ( E* x E. y ph -> E* x ph )
7		df-mo 2023	. . . . . 6 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8	6, 7	sylib 121	. . . . 5 |- ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9	4, 8	eximdh 1604	. . . 4 |- ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10	2, 9	syl5bi 151	. . 3 |- ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11	10	impcom 124	. 2 |- ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12	1, 11	sylbi 120	1 |- ( E! x E. y ph -> E. y E! x ph )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   E!weu 2019   E*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by: (None)