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Theorem drnf2 1713
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 1710 . . . 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 1710 . 2  |-  ( A. x  x  =  y  ->  ( A. z (
ph  ->  A. z ph )  <->  A. z ( ps  ->  A. z ps ) ) )
5 df-nf 1438 . 2  |-  ( F/ z ph  <->  A. z
( ph  ->  A. z ph ) )
6 df-nf 1438 . 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 1330   F/wnf 1437
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  nfsbxy  1916  nfsbxyt  1917  drnfc2  2300
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