Theorem nfald 1690

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 1457	. . 3 |- ( ph -> A. y ph )
3		nfald.2	. . 3 |- ( ph -> F/ x ps )
4	2, 3	alrimih 1403	. 2 |- ( ph -> A. y F/ x ps )
5		nfnf1 1481	. . . 4 |- F/ x F/ x ps
6	5	nfal 1513	. . 3 |- F/ x A. y F/ x ps
7		hba1 1478	. . . 4 |- ( A. y F/ x ps -> A. y A. y F/ x ps )
8		sp 1446	. . . . 5 |- ( A. y F/ x ps -> F/ x ps )
9	8	nfrd 1458	. . . 4 |- ( A. y F/ x ps -> ( ps -> A. x ps ) )
10	7, 9	hbald 1425	. . 3 |- ( A. y F/ x ps -> ( A. y ps -> A. x A. y ps ) )
11	6, 10	nfd 1461	. 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 1287   F/wnf 1394

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 1381  ax-7 1382  ax-gen 1383  ax-4 1445  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  dvelimALT  1934  dvelimfv  1935  nfeudv  1963  nfeqd  2243  nfraldxy  2410  nfiotadxy  4970  bdsepnft  11435  strcollnft  11536