Theorem sbcn1

Description: Move negation in and out of class substitution. One direction of sbcng that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcn1	|- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )

Proof

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