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Theorem drex1 2006
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 2004 . . 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 1548 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1548 . 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 1546   E.wex 1547
This theorem is referenced by:  exdistrf  2010  drsb1  2055  eujustALT  2241  copsexg  4383  dfid3  4440  dropab1  27318  dropab2  27319  e2ebind  27993  e2ebindVD  28365  e2ebindALT  28383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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