Theorem nfan1 1881

Description: A closed form of nfan 1824. (Contributed by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
nfan1.1	|- F/xph
nfan1.2	|- (ph -> F/xps)

Assertion

Ref	Expression
nfan1	|- F/x(ph /\ ps)

Proof of Theorem nfan1

Step	Hyp	Ref	Expression
1		nfan1.2	. . . . 5 |- (ph -> F/xps)
2	1	nfrd 1763	. . . 4 |- (ph -> (ps -> A.xps))
3	2	imdistani 671	. . 3 |- ((ph /\ ps) -> (ph /\ A.xps))
4		nfan1.1	. . . 4 |- F/xph
5	4	19.28 1870	. . 3 |- (A.x(ph /\ ps) <-> (ph /\ A.xps))
6	3, 5	sylibr 203	. 2 |- ((ph /\ ps) -> A.x(ph /\ ps))
7	6	nfi 1551	1 |- F/x(ph /\ ps)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  spimed  1977  ralcom2  2776  sbcralt  3119  sbcrext  3120  csbiebt  3173