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Theorem nfceqdf 2224
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfceqdf.1  |-  F/ x ph
nfceqdf.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
nfceqdf  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )

Proof of Theorem nfceqdf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfceqdf.1 . . . 4  |-  F/ x ph
2 nfceqdf.2 . . . . 5  |-  ( ph  ->  A  =  B )
32eleq2d 2154 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  y  e.  B ) )
41, 3nfbidf 1475 . . 3  |-  ( ph  ->  ( F/ x  y  e.  A  <->  F/ x  y  e.  B )
)
54albidv 1749 . 2  |-  ( ph  ->  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B ) )
6 df-nfc 2214 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
7 df-nfc 2214 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
85, 6, 73bitr4g 221 1  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285    = wceq 1287   F/wnf 1392    e. wcel 1436   F/_wnfc 2212
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cleq 2078  df-clel 2081  df-nfc 2214
This theorem is referenced by:  nfopd  3622  dfnfc2  3654  nfimad  4750  nffvd  5280
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