Theorem 2euex 1987

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 1947	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom 1554	. . . 4 |- ( E. x E. y ph <-> E. y E. x ph )
3		hbe1 1384	. . . . . 6 |- ( E. y ph -> A. y E. y ph )
4	3	hbmo 1939	. . . . 5 |- ( E* x E. y ph -> A. y E* x E. y ph )
5		19.8a 1482	. . . . . . 7 |- ( ph -> E. y ph )
6	5	moimi 1965	. . . . . 6 |- ( E* x E. y ph -> E* x ph )
7		df-mo 1904	. . . . . 6 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8	6, 7	sylib 127	. . . . 5 |- ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9	4, 8	eximdh 1502	. . . 4 |- ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10	2, 9	syl5bi 141	. . 3 |- ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11	10	impcom 116	. 2 |- ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12	1, 11	sylbi 114	1 |- ( E! x E. y ph -> E. y E! x ph )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by: (None)