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Theorem 2eu2 2393
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 2338 . . 3  |-  ( E! y E. x ph  ->  E* y E. x ph )
2 2moex 2382 . . 3  |-  ( E* y E. x ph  ->  A. x E* y ph )
3 2eu1 2392 . . . 4  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
4 simpl 463 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E. y ph )
53, 4syl6bi 236 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  E! x E. y ph ) )
61, 2, 53syl 18 . 2  |-  ( E! y E. x ph  ->  ( E! x E! y ph  ->  E! x E. y ph )
)
7 2exeu 2388 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
87expcom 441 . 2  |-  ( E! y E. x ph  ->  ( E! x E. y ph  ->  E! x E! y ph ) )
96, 8impbid 195 1  |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673   E!weu 2309   E*wmo 2310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-eu 2313  df-mo 2314
This theorem is referenced by:  2eu8  2399
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