Theorem sbcimdv 3016

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1445). (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 2959	. 2 |- ( [. A / x ]. ps -> A e. _V )
2		sbcimdv.1	. . . . 5 |- ( ph -> ( ps -> ch ) )
3	2	alrimiv 1862	. . . 4 |- ( ph -> A. x ( ps -> ch ) )
4		spsbc 2962	. . . 4 |- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
5		sbcim1 2999	. . . 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 1341   e. wcel 2136   _Vcvv 2726   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by: (None)