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Theorem drex1 2176
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 301 . . . 4  |-  ( A. x  x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32dral1 2174 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  -.  ph  <->  A. y  -.  ps )
)
43notbid 301 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. x  -.  ph  <->  -.  A. y  -.  ps ) )
5 df-ex 1672 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1672 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4g 296 1  |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  exdistrf  2182  drsb1  2226  eujustALT  2322  copsexg  4687  dfid3  4755  dropab1  36870  dropab2  36871  e2ebind  37000  e2ebindVD  37372  e2ebindALT  37389
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