Theorem sbciedf

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014)

	Ref	Expression
Hypotheses	sbcied.1	|- ( ph -> A e. V )
	sbcied.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
	sbciedf.3	|- F/ x ph
	sbciedf.4	|- ( ph -> F/ x ch )
Assertion	sbciedf	|- ( 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		sbciedf.3	|- F/ x ph
4		sbciedf.4	|- ( ph -> F/ x ch )
5	2	ex	|- ( ph -> ( x = A -> ( ps <-> ch ) ) )
6	3 5	alrimi	|- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		sbciegft	|- ( ( A e. V /\ F/ x ch /\ A. x ( x = A -> ( ps <-> ch ) ) ) -> ( [. A / x ]. ps <-> ch ) )
8	1 4 6 7	syl3anc	|- ( ph -> ( [. A / x ]. ps <-> ch ) )