Theorem 2euex 2352

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 2296	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom 1835	. . . 4 |- ( E. x E. y ph <-> E. y E. x ph )
3		nfe1 1826	. . . . . 6 |- F/ y E. y ph
4	3	nfmo 2287	. . . . 5 |- F/ y E* x E. y ph
5		19.8a 1843	. . . . . . 7 |- ( ph -> E. y ph )
6	5	moimi 2326	. . . . . 6 |- ( E* x E. y ph -> E* x ph )
7		df-mo 2273	. . . . . 6 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8	6, 7	sylib 196	. . . . 5 |- ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9	4, 8	eximd 1868	. . . 4 |- ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10	2, 9	syl5bi 217	. . 3 |- ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11	10	impcom 430	. 2 |- ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12	1, 11	sylbi 195	1 |- ( E! x E. y ph -> E. y E! x ph )

Syntax hints:   -> wi 4   /\ wa 369   E.wex 1599   E!weu 2268   E*wmo 2269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-eu 2272  df-mo 2273

This theorem is referenced by:  2exeu  2357