Theorem sbimdv

Description: Deduction substituting both sides of an implication, with ph and x disjoint. See also sbimd . (Contributed by Wolf Lammen, 6-May-2023) Revise df-sb . (Revised by Steven Nguyen, 6-Jul-2023)

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

Proof

Step	Hyp	Ref	Expression
1		sbimdv.1	|- ( ph -> ( ps -> ch ) )
2	1	alrimiv	|- ( ph -> A. x ( ps -> ch ) )
3		spsbim	|- ( A. x ( ps -> ch ) -> ( [ t / x ] ps -> [ t / x ] ch ) )
4	2 3	syl	|- ( ph -> ( [ t / x ] ps -> [ t / x ] ch ) )