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Theorem 2euex 2383
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 2335 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1937 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 nfe1 1928 . . . . . 6  |-  F/ y E. y ph
43nfmo 2326 . . . . 5  |-  F/ y E* x E. y ph
5 19.8a 1945 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2359 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 2314 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 201 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximd 1970 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 225 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 436 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 200 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   E.wex 1673   E!weu 2309   E*wmo 2310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-eu 2313  df-mo 2314
This theorem is referenced by:  2exeu  2388
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