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Theorem drex1 2055
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.)
Hypothesis
Ref Expression
dral1.1  |-  ( 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 dral1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21notbid 286 . . . 4  |-  ( A. x  x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32dral1 2053 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  -.  ph  <->  A. y  -.  ps )
)
43notbid 286 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. x  -.  ph  <->  -.  A. y  -.  ps ) )
5 df-ex 1551 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1551 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4g 280 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550
This theorem is referenced by:  exdistrf  2062  exdistrfOLD  2063  drsb1  2102  eujustALT  2283  copsexg  4429  dfid3  4486  dropab1  27559  dropab2  27560  e2ebind  28403  e2ebindVD  28776  e2ebindALT  28794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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