Theorem sbc2iedv

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Proof shortened by Mario Carneiro, 18-Oct-2016)

	Ref	Expression
Hypotheses	sbc2iedv.1	|- A e. _V
	sbc2iedv.2	|- B e. _V
	sbc2iedv.3	|- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )
Assertion	sbc2iedv	|- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbc2iedv.1	|- A e. _V
2		sbc2iedv.2	|- B e. _V
3		sbc2iedv.3	|- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )
4	1	a1i	|- ( ph -> A e. _V )
5	2	a1i	|- ( ( ph /\ x = A ) -> B e. _V )
6	3	impl	|- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ps <-> ch ) )
7	5 6	sbcied	|- ( ( ph /\ x = A ) -> ( [. B / y ]. ps <-> ch ) )
8	4 7	sbcied	|- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )