Theorem sbcied

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypotheses	sbcied.1	|- ( ph -> A e. V )
	sbcied.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
Assertion	sbcied	|- ( ph -> ( [. A / x ]. ps <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	|- ( ph -> A e. V )
2		sbcied.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		df-sbc	|- ( [. A / x ]. ps <-> A e. { x | ps } )
4	1 2	elabd3	|- ( ph -> ( A e. { x | ps } <-> ch ) )
5	3 4	bitrid	|- ( ph -> ( [. A / x ]. ps <-> ch ) )