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Theorem 2euex 2029
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 1989 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1595 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 hbe1 1425 . . . . . 6  |-  ( E. y ph  ->  A. y E. y ph )
43hbmo 1981 . . . . 5  |-  ( E* x E. y ph  ->  A. y E* x E. y ph )
5 19.8a 1523 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2007 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 1946 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 120 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximdh 1543 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 150 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 123 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 119 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1422   E!weu 1942   E*wmo 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by: (None)
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