Theorem nfald 1806

Description: If x is not free in ph, it is not free in A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)

Hypotheses

Ref	Expression
nfald.1	|- F/ y ph
nfald.2	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfald	|- ( ph -> F/ x A. y ps )

Proof of Theorem nfald

Step	Hyp	Ref	Expression
1		nfald.1	. . . 4 |- F/ y ph
2	1	nfri 1565	. . 3 |- ( ph -> A. y ph )
3		nfald.2	. . 3 |- ( ph -> F/ x ps )
4	2, 3	alrimih 1515	. 2 |- ( ph -> A. y F/ x ps )
5		nfnf1 1590	. . . 4 |- F/ x F/ x ps
6	5	nfal 1622	. . 3 |- F/ x A. y F/ x ps
7		hba1 1586	. . . 4 |- ( A. y F/ x ps -> A. y A. y F/ x ps )
8		sp 1557	. . . . 5 |- ( A. y F/ x ps -> F/ x ps )
9	8	nfrd 1566	. . . 4 |- ( A. y F/ x ps -> ( ps -> A. x ps ) )
10	7, 9	hbald 1537	. . 3 |- ( A. y F/ x ps -> ( A. y ps -> A. x A. y ps ) )
11	6, 10	nfd 1569	. 2 |- ( A. y F/ x ps -> F/ x A. y ps )
12	4, 11	syl 14	1 |- ( ph -> F/ x A. y ps )

Syntax hints:   -> wi 4   A.wal 1393   F/wnf 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  dvelimALT  2061  dvelimfv  2062  nfeudv  2092  nfeqd  2387  nfraldw  2562  nfraldxy  2563  nfiotadw  5281  nfixpxy  6864  bdsepnft  16250