Theorem nfimd 1599

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 1558	. . . . 5 |- F/ x F/ x ps
4	3	nfri 1533	. . . 4 |- ( F/ x ps -> A. x F/ x ps )
5		nfnf1 1558	. . . . 5 |- F/ x F/ x ch
6	5	nfri 1533	. . . 4 |- ( F/ x ch -> A. x F/ x ch )
7		nfr 1532	. . . . . 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 1596	. . . . . 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 1493	. . 3 |- ( F/ x ps -> ( F/ x ch -> A. x ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) )
13		df-nf 1475	. . 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 1362   F/wnf 1474

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 1461  ax-gen 1463  ax-4 1524  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nfbid  1602  dvelimALT  2029  dvelimfv  2030  dvelimor  2037  nfmod  2062  nfraldw  2529  nfraldxy  2530  nfixpxy  6776  cbvrald  15434