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Theorem nfan1 1501
Description: A closed form of nfan 1502. (Contributed by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
nfan1.1  |-  F/ x ph
nfan1.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfan1  |-  F/ x
( ph  /\  ps )

Proof of Theorem nfan1
StepHypRef Expression
1 nfan1.2 . . . . 5  |-  ( ph  ->  F/ x ps )
21nfrd 1458 . . . 4  |-  ( ph  ->  ( ps  ->  A. x ps ) )
32imdistani 434 . . 3  |-  ( (
ph  /\  ps )  ->  ( ph  /\  A. x ps ) )
4 nfan1.1 . . . . 5  |-  F/ x ph
54nfri 1457 . . . 4  |-  ( ph  ->  A. x ph )
6519.28h 1499 . . 3  |-  ( A. x ( ph  /\  ps )  <->  ( ph  /\  A. x ps ) )
73, 6sylibr 132 . 2  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
87nfi 1396 1  |-  F/ x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287   F/wnf 1394
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-gen 1383  ax-4 1445
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  nfan  1502  sbcralt  2913  sbcrext  2914  csbiebt  2965  riota5f  5614
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