Theorem nfald 1783

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 1542	. . 3 |- ( ph -> A. y ph )
3		nfald.2	. . 3 |- ( ph -> F/ x ps )
4	2, 3	alrimih 1492	. 2 |- ( ph -> A. y F/ x ps )
5		nfnf1 1567	. . . 4 |- F/ x F/ x ps
6	5	nfal 1599	. . 3 |- F/ x A. y F/ x ps
7		hba1 1563	. . . 4 |- ( A. y F/ x ps -> A. y A. y F/ x ps )
8		sp 1534	. . . . 5 |- ( A. y F/ x ps -> F/ x ps )
9	8	nfrd 1543	. . . 4 |- ( A. y F/ x ps -> ( ps -> A. x ps ) )
10	7, 9	hbald 1514	. . 3 |- ( A. y F/ x ps -> ( A. y ps -> A. x A. y ps ) )
11	6, 10	nfd 1546	. 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 1371   F/wnf 1483

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 1470  ax-7 1471  ax-gen 1472  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  dvelimALT  2038  dvelimfv  2039  nfeudv  2069  nfeqd  2363  nfraldw  2538  nfraldxy  2539  nfiotadw  5235  nfixpxy  6804  bdsepnft  15827