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Theorem 2euex 2355
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
StepHypRef Expression
1 eu5 2321 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1757 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 nfe1 1748 . . . . . 6  |-  F/ y E. y ph
43nfmo 2300 . . . . 5  |-  F/ y E* x E. y ph
5 19.8a 1763 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2330 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 2288 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 190 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximd 1787 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 210 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 421 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 189 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551   E!weu 2283   E*wmo 2284
This theorem is referenced by:  2exeu  2360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288
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