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Theorem nfceqi 2216
Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfceqi.1  |-  A  =  B
Assertion
Ref Expression
nfceqi  |-  ( F/_ x A  <->  F/_ x B )

Proof of Theorem nfceqi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfceqi.1 . . . . 5  |-  A  =  B
21eleq2i 2146 . . . 4  |-  ( y  e.  A  <->  y  e.  B )
32nfbii 1403 . . 3  |-  ( F/ x  y  e.  A  <->  F/ x  y  e.  B
)
43albii 1400 . 2  |-  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B )
5 df-nfc 2209 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
6 df-nfc 2209 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
74, 5, 63bitr4i 210 1  |-  ( F/_ x A  <->  F/_ x B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283    = wceq 1285   F/wnf 1390    e. wcel 1434   F/_wnfc 2207
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2075  df-clel 2078  df-nfc 2209
This theorem is referenced by:  nfcxfr  2217  nfcxfrd  2218
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