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Theorem 2euswap 2221
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 1787 . . . 4  |-  ( E. x E. y ph  ->  E. y E. x ph )
21a1i 10 . . 3  |-  ( A. x E* y ph  ->  ( E. x E. y ph  ->  E. y E. x ph ) )
3 2moswap 2220 . . 3  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
42, 3anim12d 546 . 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 2183 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
6 eu5 2183 . 2  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
74, 5, 63imtr4g 261 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 358   A.wal 1529   E.wex 1530   E!weu 2145   E*wmo 2146
This theorem is referenced by:  euxfr2  2952  2reuswap  2969  2reuswap2  23139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150
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