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Theorem nfan1 1543
Description: A closed form of nfan 1544. (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 1500 . . . 4  |-  ( ph  ->  ( ps  ->  A. x ps ) )
32imdistani 441 . . 3  |-  ( (
ph  /\  ps )  ->  ( ph  /\  A. x ps ) )
4 nfan1.1 . . . . 5  |-  F/ x ph
54nfri 1499 . . . 4  |-  ( ph  ->  A. x ph )
6519.28h 1541 . . 3  |-  ( A. x ( ph  /\  ps )  <->  ( ph  /\  A. x ps ) )
73, 6sylibr 133 . 2  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
87nfi 1438 1  |-  F/ x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1329   F/wnf 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  nfan  1544  sbcralt  2980  sbcrext  2981  csbiebt  3034  riota5f  5747
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