Theorem sbcimdv 3498

Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1738). (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 3445	. 2 |- ( [. A / x ]. ps -> A e. _V )
2		sbcimdv.1	. . . . 5 |- ( ph -> ( ps -> ch ) )
3	2	alrimiv 1855	. . . 4 |- ( ph -> A. x ( ps -> ch ) )
4		spsbc 3448	. . . 4 |- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
5		sbcim1 3482	. . . 4 |- ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
6	3, 4, 5	syl56 36	. . 3 |- ( A e. _V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7	6	com3l 89	. 2 |- ( ph -> ( [. A / x ]. ps -> ( A e. _V -> [. A / x ]. ch ) ) )
8	1, 7	mpdi 45	1 |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  esum2dlem  30154