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Theorem nfceqi 2415
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 2347 . . . 4  |-  ( y  e.  A  <->  y  e.  B )
32nfbii 1556 . . 3  |-  ( F/ x  y  e.  A  <->  F/ x  y  e.  B
)
43albii 1553 . 2  |-  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B )
5 df-nfc 2408 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
6 df-nfc 2408 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
74, 5, 63bitr4i 268 1  |-  ( F/_ x A  <->  F/_ x B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406
This theorem is referenced by:  nfcxfr  2416  nfcxfrd  2417  ballotlem7  23094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-nfc 2408
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