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Theorem drnfc2 2519
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
drnfc1.1  |-  ( A. x  x  =  y  ->  A  =  B )
Assertion
Ref Expression
drnfc2  |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )

Proof of Theorem drnfc2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5  |-  ( A. x  x  =  y  ->  A  =  B )
21eleq2d 2433 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf2 1983 . . 3  |-  ( A. x  x  =  y  ->  ( F/ z  w  e.  A  <->  F/ z  w  e.  B )
)
43dral2 1979 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ z  w  e.  A  <->  A. w F/ z  w  e.  B ) )
5 df-nfc 2491 . 2  |-  ( F/_ z A  <->  A. w F/ z  w  e.  A )
6 df-nfc 2491 . 2  |-  ( F/_ z B  <->  A. w F/ z  w  e.  B )
74, 5, 63bitr4g 279 1  |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1545   F/wnf 1549    = wceq 1647    e. wcel 1715   F/_wnfc 2489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1547  df-nf 1550  df-cleq 2359  df-clel 2362  df-nfc 2491
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