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Theorem 2eu7 2229
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 1706 . . . 4  |-  F/ x E. x ph
21nfeu 2159 . . 3  |-  F/ x E! y E. x ph
32euan 2200 . 2  |-  ( E! x ( E! y E. x ph  /\  E. y ph )  <->  ( E! y E. x ph  /\  E! x E. y ph ) )
4 ancom 437 . . . . 5  |-  ( ( E. x ph  /\  E. y ph )  <->  ( E. y ph  /\  E. x ph ) )
54eubii 2152 . . . 4  |-  ( E! y ( E. x ph  /\  E. y ph ) 
<->  E! y ( E. y ph  /\  E. x ph ) )
6 nfe1 1706 . . . . 5  |-  F/ y E. y ph
76euan 2200 . . . 4  |-  ( E! y ( E. y ph  /\  E. x ph ) 
<->  ( E. y ph  /\  E! y E. x ph ) )
8 ancom 437 . . . 4  |-  ( ( E. y ph  /\  E! y E. x ph ) 
<->  ( E! y E. x ph  /\  E. y ph ) )
95, 7, 83bitri 262 . . 3  |-  ( E! y ( E. x ph  /\  E. y ph ) 
<->  ( E! y E. x ph  /\  E. y ph ) )
109eubii 2152 . 2  |-  ( E! x E! y ( E. x ph  /\  E. y ph )  <->  E! x
( E! y E. x ph  /\  E. y ph ) )
11 ancom 437 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E! y E. x ph  /\  E! x E. y ph ) )
123, 10, 113bitr4ri 269 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 set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   E!weu 2143
This theorem is referenced by:  2eu8  2230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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