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Theorem drex1 2029
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 294 . . . 4  |-  ( A. x  x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32dral1 2027 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  -.  ph  <->  A. y  -.  ps )
)
43notbid 294 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. x  -.  ph  <->  -.  A. y  -.  ps ) )
5 df-ex 1588 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1588 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4g 288 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  exdistrf  2035  drsb1  2078  eujustALT  2265  copsexg  4685  copsexgOLD  4686  dfid3  4746  dropab1  29852  dropab2  29853  e2ebind  31605  e2ebindVD  31981  e2ebindALT  31998
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