Theorem spsbim

Description: Distribute substitution over implication. Closed form of sbimi . Specialization of implication. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Steven Nguyen, 24-Jul-2023)

	Ref	Expression
Assertion	spsbim	|- ( A. x ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) )

Proof

Step	Hyp	Ref	Expression
1		stdpc4	|- ( A. x ( ph -> ps ) -> [ t / x ] ( ph -> ps ) )
2		sbi1	|- ( [ t / x ] ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) )
3	1 2	syl	|- ( A. x ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) )