Theorem sbcimdv 3339

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

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 1686	. . 3 |- ( ph -> A. x ( ps -> ch ) )
3		spsbc 3283	. . 3 |- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
4		sbcim1 3319	. . 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 3280	. . . . 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 1368   e. wcel 1757   _Vcvv 3054   [.wsbc 3270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-12 1793  ax-13 1944  ax-ext 2429

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-v 3056  df-sbc 3271

This theorem is referenced by: (None)