Theorem sbcimdv 3395

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90 (alim 1633). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.)

Hypothesis

Ref	Expression
sbcimdv.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
sbcimdv	|- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem sbcimdv

Step	Hyp	Ref	Expression
1		sbcimdv.1	. . . 4 |- ( ph -> ( ps -> ch ) )
2	1	alrimiv 1720	. . 3 |- ( ph -> A. x ( ps -> ch ) )
3		spsbc 3340	. . 3 |- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
4		sbcim1 3376	. . 3 |- ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
5	2, 3, 4	syl56 34	. 2 |- ( A e. _V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
6		sbcex 3337	. . . . 5 |- ( [. A / x ]. ps -> A e. _V )
7	6	con3i 135	. . . 4 |- ( -. A e. _V -> -. [. A / x ]. ps )
8	7	pm2.21d 106	. . 3 |- ( -. A e. _V -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
9	8	a1d 25	. 2 |- ( -. A e. _V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
10	5, 9	pm2.61i 164	1 |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1393   e. wcel 1819   _Vcvv 3109   [.wsbc 3327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328

This theorem is referenced by:  esum2dlem  28264