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Theorem drnfc2 2246
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 2157 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf2 1669 . . 3  |-  ( A. x  x  =  y  ->  ( F/ z  w  e.  A  <->  F/ z  w  e.  B )
)
43dral2 1666 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ z  w  e.  A  <->  A. w F/ z  w  e.  B ) )
5 df-nfc 2217 . 2  |-  ( F/_ z A  <->  A. w F/ z  w  e.  A )
6 df-nfc 2217 . 2  |-  ( F/_ z B  <->  A. w F/ z  w  e.  B )
74, 5, 63bitr4g 221 1  |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394    e. wcel 1438   F/_wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-nfc 2217
This theorem is referenced by: (None)
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