Theorem nfimd 1608

Description: If in a context x is not free in ps and ch, then it is not free in ( ps -> ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypotheses

Ref	Expression
nfimd.1	|- ( ph -> F/ x ps )
nfimd.2	|- ( ph -> F/ x ch )

Assertion

Ref	Expression
nfimd	|- ( ph -> F/ x ( ps -> ch ) )

Proof of Theorem nfimd

Step	Hyp	Ref	Expression
1		nfimd.1	. 2 |- ( ph -> F/ x ps )
2		nfimd.2	. 2 |- ( ph -> F/ x ch )
3		nfnf1 1567	. . . . 5 |- F/ x F/ x ps
4	3	nfri 1542	. . . 4 |- ( F/ x ps -> A. x F/ x ps )
5		nfnf1 1567	. . . . 5 |- F/ x F/ x ch
6	5	nfri 1542	. . . 4 |- ( F/ x ch -> A. x F/ x ch )
7		nfr 1541	. . . . . 6 |- ( F/ x ch -> ( ch -> A. x ch ) )
8	7	imim2d 54	. . . . 5 |- ( F/ x ch -> ( ( ps -> ch ) -> ( ps -> A. x ch ) ) )
9		19.21t 1605	. . . . . 6 |- ( F/ x ps -> ( A. x ( ps -> ch ) <-> ( ps -> A. x ch ) ) )
10	9	biimprd 158	. . . . 5 |- ( F/ x ps -> ( ( ps -> A. x ch ) -> A. x ( ps -> ch ) ) )
11	8, 10	syl9r 73	. . . 4 |- ( F/ x ps -> ( F/ x ch -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) )
12	4, 6, 11	alrimdh 1502	. . 3 |- ( F/ x ps -> ( F/ x ch -> A. x ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) )
13		df-nf 1484	. . 3 |- ( F/ x ( ps -> ch ) <-> A. x ( ( ps -> ch ) -> A. x ( ps -> ch ) ) )
14	12, 13	imbitrrdi 162	. 2 |- ( F/ x ps -> ( F/ x ch -> F/ x ( ps -> ch ) ) )
15	1, 2, 14	sylc 62	1 |- ( ph -> F/ x ( ps -> ch ) )

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-gen 1472  ax-4 1533  ax-ial 1557  ax-i5r 1558

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

This theorem is referenced by:  nfbid  1611  dvelimALT  2038  dvelimfv  2039  dvelimor  2046  nfmod  2071  nfraldw  2538  nfraldxy  2539  nfixpxy  6806  cbvrald  15761