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Theorem 2eu7 2382
Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
Assertion
Ref Expression
2eu7  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )

Proof of Theorem 2eu7
StepHypRef Expression
1 nfe1 1780 . . . 4  |-  F/ x E. x ph
21nfeu 2281 . . 3  |-  F/ x E! y E. x ph
32euan 2340 . 2  |-  ( E! x ( E! y E. x ph  /\  E. y ph )  <->  ( E! y E. x ph  /\  E! x E. y ph ) )
4 ancom 450 . . . . 5  |-  ( ( E. x ph  /\  E. y ph )  <->  ( E. y ph  /\  E. x ph ) )
54eubii 2287 . . . 4  |-  ( E! y ( E. x ph  /\  E. y ph ) 
<->  E! y ( E. y ph  /\  E. x ph ) )
6 nfe1 1780 . . . . 5  |-  F/ y E. y ph
76euan 2340 . . . 4  |-  ( E! y ( E. y ph  /\  E. x ph ) 
<->  ( E. y ph  /\  E! y E. x ph ) )
8 ancom 450 . . . 4  |-  ( ( E. y ph  /\  E! y E. x ph ) 
<->  ( E! y E. x ph  /\  E. y ph ) )
95, 7, 83bitri 271 . . 3  |-  ( E! y ( E. x ph  /\  E. y ph ) 
<->  ( E! y E. x ph  /\  E. y ph ) )
109eubii 2287 . 2  |-  ( E! x E! y ( E. x ph  /\  E. y ph )  <->  E! x
( E! y E. x ph  /\  E. y ph ) )
11 ancom 450 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E! y E. x ph  /\  E! x E. y ph ) )
123, 10, 113bitr4ri 278 1  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587   E!weu 2262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-eu 2266
This theorem is referenced by:  2eu8  2383
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