Theorem 2euex 2062

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 2022	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom 1625	. . . 4 |- ( E. x E. y ph <-> E. y E. x ph )
3		hbe1 1454	. . . . . 6 |- ( E. y ph -> A. y E. y ph )
4	3	hbmo 2014	. . . . 5 |- ( E* x E. y ph -> A. y E* x E. y ph )
5		19.8a 1552	. . . . . . 7 |- ( ph -> E. y ph )
6	5	moimi 2040	. . . . . 6 |- ( E* x E. y ph -> E* x ph )
7		df-mo 1979	. . . . . 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 1573	. . . 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 1451   E!weu 1975   E*wmo 1976

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979

This theorem is referenced by: (None)