Theorem sbcimdv 3071

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1481). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)

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		sbcex 3014	. 2 |- ( [. A / x ]. ps -> A e. _V )
2		sbcimdv.1	. . . . 5 |- ( ph -> ( ps -> ch ) )
3	2	alrimiv 1898	. . . 4 |- ( ph -> A. x ( ps -> ch ) )
4		spsbc 3017	. . . 4 |- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
5		sbcim1 3054	. . . 4 |- ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
6	3, 4, 5	syl56 34	. . 3 |- ( A e. _V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7	6	com3l 81	. 2 |- ( ph -> ( [. A / x ]. ps -> ( A e. _V -> [. A / x ]. ch ) ) )
8	1, 7	mpdi 43	1 |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   A.wal 1371   e. wcel 2178   _Vcvv 2776   [.wsbc 3005

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006

This theorem is referenced by: (None)