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Theorem drnf2 1697
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
drex2.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
drnf2  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps ) )

Proof of Theorem drnf2
StepHypRef Expression
1 drex2.1 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral2 1694 . . . 4  |-  ( A. x  x  =  y  ->  ( A. z ph  <->  A. z ps ) )
31, 2imbi12d 233 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. z ph )  <->  ( ps  ->  A. z ps )
) )
43dral2 1694 . 2  |-  ( A. x  x  =  y  ->  ( A. z (
ph  ->  A. z ph )  <->  A. z ( ps  ->  A. z ps ) ) )
5 df-nf 1422 . 2  |-  ( F/ z ph  <->  A. z
( ph  ->  A. z ph ) )
6 df-nf 1422 . 2  |-  ( F/ z ps  <->  A. z
( ps  ->  A. z ps ) )
74, 5, 63bitr4g 222 1  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314   F/wnf 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  nfsbxy  1895  nfsbxyt  1896  drnfc2  2276
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