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Theorem 2euswap 2356
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2euswap  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 1757 . . . 4  |-  ( E. x E. y ph  ->  E. y E. x ph )
21a1i 11 . . 3  |-  ( A. x E* y ph  ->  ( E. x E. y ph  ->  E. y E. x ph ) )
3 2moswap 2355 . . 3  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
42, 3anim12d 547 . 2  |-  ( A. x E* y ph  ->  ( ( E. x E. y ph  /\  E* x E. y ph )  -> 
( E. y E. x ph  /\  E* y E. x ph )
) )
5 eu5 2318 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
6 eu5 2318 . 2  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
74, 5, 63imtr4g 262 1  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   E!weu 2280   E*wmo 2281
This theorem is referenced by:  euxfr2  3111  2reuswap  3128  2reuswap2  23967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285
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