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Theorem drex1 1752
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.)
Hypothesis
Ref Expression
drex1.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
drex1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )

Proof of Theorem drex1
StepHypRef Expression
1 hbae 1679 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 drex1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
3 ax-4 1470 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
43biantrurd 301 . . . . 5  |-  ( A. x  x  =  y  ->  ( ps  <->  ( x  =  y  /\  ps )
) )
52, 4bitr2d 188 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  y  /\  ps )  <->  ph ) )
61, 5exbidh 1576 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  <->  E. x ph )
)
7 ax11e 1750 . . . 4  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps )
)
87sps 1500 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps ) )
96, 8sylbird 169 . 2  |-  ( A. x  x  =  y  ->  ( E. x ph  ->  E. y ps )
)
10 hbae 1679 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
11 equcomi 1663 . . . . . . 7  |-  ( x  =  y  ->  y  =  x )
1211sps 1500 . . . . . 6  |-  ( A. x  x  =  y  ->  y  =  x )
1312biantrurd 301 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ( y  =  x  /\  ph )
) )
1413, 2bitr3d 189 . . . 4  |-  ( A. x  x  =  y  ->  ( ( y  =  x  /\  ph )  <->  ps ) )
1510, 14exbidh 1576 . . 3  |-  ( A. x  x  =  y  ->  ( E. y ( y  =  x  /\  ph )  <->  E. y ps )
)
16 ax11e 1750 . . . . 5  |-  ( y  =  x  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph )
)
1716sps 1500 . . . 4  |-  ( A. y  y  =  x  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph ) )
1817alequcoms 1479 . . 3  |-  ( A. x  x  =  y  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph ) )
1915, 18sylbird 169 . 2  |-  ( A. x  x  =  y  ->  ( E. y ps 
->  E. x ph )
)
209, 19impbid 128 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314   E.wex 1451
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  drsb1  1753  exdistrfor  1754  copsexg  4134
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