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Theorem 2euex 2064
Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex  |-  ( E! x E. y ph  ->  E. y E! x ph )

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2024 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1627 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 hbe1 1456 . . . . . 6  |-  ( E. y ph  ->  A. y E. y ph )
43hbmo 2016 . . . . 5  |-  ( E* x E. y ph  ->  A. y E* x E. y ph )
5 19.8a 1554 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2042 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 1981 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 121 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximdh 1575 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 151 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 124 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 120 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1453   E!weu 1977   E*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by: (None)
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