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Theorem drnf1 1721
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
drnf1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )

Proof of Theorem drnf1
StepHypRef Expression
1 drex2.1 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral1 1718 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
31, 2imbi12d 233 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ph )  <->  ( ps  ->  A. y ps )
) )
43dral1 1718 . 2  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  A. x ph )  <->  A. y ( ps  ->  A. y ps ) ) )
5 df-nf 1449 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
6 df-nf 1449 . 2  |-  ( F/ y ps  <->  A. y
( ps  ->  A. y ps ) )
74, 5, 63bitr4g 222 1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1341   F/wnf 1448
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 1435  ax-7 1436  ax-gen 1437  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522
This theorem depends on definitions:  df-bi 116  df-nf 1449
This theorem is referenced by:  drnfc1  2325
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