Theorem sbcied2

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014)

	Ref	Expression
Hypotheses	sbcied2.1	|- ( ph -> A e. V )
	sbcied2.2	|- ( ph -> A = B )
	sbcied2.3	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
Assertion	sbcied2	|- ( ph -> ( [. A / x ]. ps <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbcied2.1	|- ( ph -> A e. V )
2		sbcied2.2	|- ( ph -> A = B )
3		sbcied2.3	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
4		id	|- ( x = A -> x = A )
5	4 2	sylan9eqr	|- ( ( ph /\ x = A ) -> x = B )
6	5 3	syldan	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
7	1 6	sbcied	|- ( ph -> ( [. A / x ]. ps <-> ch ) )