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Theorem nfceqi 2568
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 2500 . . . 4  |-  ( y  e.  A  <->  y  e.  B )
32nfbii 1578 . . 3  |-  ( F/ x  y  e.  A  <->  F/ x  y  e.  B
)
43albii 1575 . 2  |-  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B )
5 df-nfc 2561 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
6 df-nfc 2561 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
74, 5, 63bitr4i 269 1  |-  ( F/_ x A  <->  F/_ x B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559
This theorem is referenced by:  nfcxfr  2569  nfcxfrd  2570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2429  df-clel 2432  df-nfc 2561
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