Theorem 2euex 2174

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 2134	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom 1716	. . . 4 |- ( E. x E. y ph <-> E. y E. x ph )
3		hbe1 1548	. . . . . 6 |- ( E. y ph -> A. y E. y ph )
4	3	hbmo 2125	. . . . 5 |- ( E* x E. y ph -> A. y E* x E. y ph )
5		19.8a 1643	. . . . . . 7 |- ( ph -> E. y ph )
6	5	moimi 2152	. . . . . 6 |- ( E* x E. y ph -> E* x ph )
7		df-mo 2090	. . . . . 6 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8	6, 7	sylib 122	. . . . 5 |- ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9	4, 8	eximdh 1664	. . . 4 |- ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10	2, 9	biimtrid 152	. . 3 |- ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11	10	impcom 125	. 2 |- ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12	1, 11	sylbi 121	1 |- ( E! x E. y ph -> E. y E! x ph )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1545   E!weu 2086   E*wmo 2087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)