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Theorem 2eu7 2395
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 1789 . . . 4  |-  F/ x E. x ph
21nfeu 2294 . . 3  |-  F/ x E! y E. x ph
32euan 2353 . 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 2300 . . . 4  |-  ( E! y ( E. x ph  /\  E. y ph ) 
<->  E! y ( E. y ph  /\  E. x ph ) )
6 nfe1 1789 . . . . 5  |-  F/ y E. y ph
76euan 2353 . . . 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 2300 . 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 1596   E!weu 2275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-eu 2279
This theorem is referenced by:  2eu8  2396
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