Theorem sbcim1

Description: Distribution of class substitution over implication. One direction of sbcimg that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcim1	|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )

Proof

Step	Hyp	Ref	Expression
1		sbcex	|- ( [. A / x ]. ( ph -> ps ) -> A e. _V )
2		sbcimg	|- ( A e. _V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
3	2	biimpd	|- ( A e. _V -> ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
4	1 3	mpcom	|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )