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Theorem nfan1 1497
Description: A closed form of nfan 1498. (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 1454 . . . 4  |-  ( ph  ->  ( ps  ->  A. x ps ) )
32imdistani 434 . . 3  |-  ( (
ph  /\  ps )  ->  ( ph  /\  A. x ps ) )
4 nfan1.1 . . . . 5  |-  F/ x ph
54nfri 1453 . . . 4  |-  ( ph  ->  A. x ph )
6519.28h 1495 . . 3  |-  ( A. x ( ph  /\  ps )  <->  ( ph  /\  A. x ps ) )
73, 6sylibr 132 . 2  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
87nfi 1392 1  |-  F/ x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   F/wnf 1390
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 1377  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfan  1498  sbcralt  2891  sbcrext  2892  csbiebt  2943  riota5f  5523
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