Theorem sbcbi1

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

	Ref	Expression
Assertion	sbcbi1	|- ( [. 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		sbcbig	|- ( 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 ) )