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Theorem nfeqd 2243
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1  |-  ( ph  -> 
F/_ x A )
nfeqd.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfeqd  |-  ( ph  ->  F/ x  A  =  B )

Proof of Theorem nfeqd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2082 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1466 . . 3  |-  F/ y
ph
3 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
43nfcrd 2242 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
5 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
65nfcrd 2242 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
74, 6nfbid 1525 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  <->  y  e.  B ) )
82, 7nfald 1690 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  <->  y  e.  B ) )
91, 8nfxfrd 1409 1  |-  ( ph  ->  F/ x  A  =  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-5 1381  ax-7 1382  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-nfc 2217
This theorem is referenced by:  nfeld  2244  nfned  2349  vtoclgft  2669  sbcralt  2915  sbcrext  2916  csbiebt  2967  dfnfc2  3671  eusvnfb  4276  eusv2i  4277  iota2df  5004  riota5f  5632
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