Theorem sbcimdv

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

	Ref	Expression
Hypothesis	sbcimdv.1	|- ( ph -> ( ps -> ch ) )
Assertion	sbcimdv	|- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbcimdv.1	|- ( ph -> ( ps -> ch ) )
2		sbcex	|- ( [. A / x ]. ps -> A e. _V )
3	1	alrimiv	|- ( ph -> A. x ( ps -> ch ) )
4		spsbc	|- ( A e. _V -> ( A. x ( ps -> ch ) -> [. A / x ]. ( ps -> ch ) ) )
5		sbcim1	|- ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
6	3 4 5	syl56	|- ( A e. _V -> ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7	6	com3l	|- ( ph -> ( [. A / x ]. ps -> ( A e. _V -> [. A / x ]. ch ) ) )
8	2 7	mpdi	|- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) )