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Theorem 2euswap 2367
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 1855 . . . 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 2366 . . 3  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
42, 3anim12d 561 . 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 2312 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
6 eu5 2312 . 2  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
74, 5, 63imtr4g 270 1  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396   E.wex 1617   E!weu 2284   E*wmo 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-eu 2288  df-mo 2289
This theorem is referenced by:  2eu1  2373  euxfr2  3281  2reuswap  3299  2reuswap2  27585
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