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Theorem drnf1 2127
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
dral1.1  |-  ( 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 dral1.1 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral1 2123 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
31, 2imbi12d 321 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ph )  <->  ( ps  ->  A. y ps )
) )
43dral1 2123 . 2  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  A. x ph )  <->  A. y ( ps  ->  A. y ps ) ) )
5 df-nf 1664 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
6 df-nf 1664 . 2  |-  ( F/ y ps  <->  A. y
( ps  ->  A. y ps ) )
74, 5, 63bitr4g 291 1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  nfald2  2129  drnfc1  2610  wl-nfs1t  31578
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