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Theorem drex1 1726
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 1653 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 drex1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
3 ax-4 1445 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
43biantrurd 299 . . . . 5  |-  ( A. x  x  =  y  ->  ( ps  <->  ( x  =  y  /\  ps )
) )
52, 4bitr2d 187 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  y  /\  ps )  <->  ph ) )
61, 5exbidh 1550 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  <->  E. x ph )
)
7 ax11e 1724 . . . 4  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps )
)
87sps 1475 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps ) )
96, 8sylbird 168 . 2  |-  ( A. x  x  =  y  ->  ( E. x ph  ->  E. y ps )
)
10 hbae 1653 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
11 equcomi 1637 . . . . . . 7  |-  ( x  =  y  ->  y  =  x )
1211sps 1475 . . . . . 6  |-  ( A. x  x  =  y  ->  y  =  x )
1312biantrurd 299 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ( y  =  x  /\  ph )
) )
1413, 2bitr3d 188 . . . 4  |-  ( A. x  x  =  y  ->  ( ( y  =  x  /\  ph )  <->  ps ) )
1510, 14exbidh 1550 . . 3  |-  ( A. x  x  =  y  ->  ( E. y ( y  =  x  /\  ph )  <->  E. y ps )
)
16 ax11e 1724 . . . . 5  |-  ( y  =  x  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph )
)
1716sps 1475 . . . 4  |-  ( A. y  y  =  x  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph ) )
1817alequcoms 1454 . . 3  |-  ( A. x  x  =  y  ->  ( E. y ( y  =  x  /\  ph )  ->  E. x ph ) )
1915, 18sylbird 168 . 2  |-  ( A. x  x  =  y  ->  ( E. y ps 
->  E. x ph )
)
209, 19impbid 127 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  drsb1  1727  exdistrfor  1728  copsexg  4062
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