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

Step	Hyp	Ref	Expression
1		nfan1.2	. . . . 5 |- ( ph -> F/ x ps )
2	1	nfrd 1500	. . . 4 |- ( ph -> ( ps -> A. x ps ) )
3	2	imdistani 441	. . 3 |- ( ( ph /\ ps ) -> ( ph /\ A. x ps ) )
4		nfan1.1	. . . . 5 |- F/ x ph
5	4	nfri 1499	. . . 4 |- ( ph -> A. x ph )
6	5	19.28h 1541	. . 3 |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )
7	3, 6	sylibr 133	. 2 |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )
8	7	nfi 1438	1 |- F/ x ( ph /\ ps )

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  2985  sbcrext  2986  csbiebt  3039  riota5f  5754