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Theorem drnf1 1637
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 1634 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
31, 2imbi12d 227 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ph )  <->  ( ps  ->  A. y ps )
) )
43dral1 1634 . 2  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  A. x ph )  <->  A. y ( ps  ->  A. y ps ) ) )
5 df-nf 1366 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
6 df-nf 1366 . 2  |-  ( F/ y ps  <->  A. y
( ps  ->  A. y ps ) )
74, 5, 63bitr4g 216 1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257   F/wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  drnfc1  2210
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