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Theorem drex1 1859
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 287 . . . 4  |-  ( A. x  x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32dral1 1855 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  -.  ph  <->  A. y  -.  ps )
)
43notbid 287 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. x  -.  ph  <->  -.  A. y  -.  ps ) )
5 df-ex 1538 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1538 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4g 281 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1532   E.wex 1537
This theorem is referenced by:  exdistrf  1863  drsb1  1886  eujustALT  2117  copsexg  4145  dfid3  4203  dropab1  26817  dropab2  26818  e2ebind  27022  e2ebindVD  27378  e2ebindALT  27396
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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