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Theorem 2eu2 2362
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2  |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )

Proof of Theorem 2eu2
StepHypRef Expression
1 eumo 2321 . . 3  |-  ( E! y E. x ph  ->  E* y E. x ph )
2 2moex 2352 . . 3  |-  ( E* y E. x ph  ->  A. x E* y ph )
3 2eu1 2361 . . . 4  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
4 simpl 444 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E. y ph )
53, 4syl6bi 220 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  E! x E. y ph ) )
61, 2, 53syl 19 . 2  |-  ( E! y E. x ph  ->  ( E! x E! y ph  ->  E! x E. y ph )
)
7 2exeu 2358 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
87expcom 425 . 2  |-  ( E! y E. x ph  ->  ( E! x E. y ph  ->  E! x E! y ph ) )
96, 8impbid 184 1  |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   E!weu 2281   E*wmo 2282
This theorem is referenced by:  2eu8  2368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286
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