Theorem sbc2ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	sbc2ie.1	|- A e. _V
	sbc2ie.2	|- B e. _V
	sbc2ie.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
Assertion	sbc2ie	|- ( [. A / x ]. [. B / y ]. ph <-> ps )

Proof

Step	Hyp	Ref	Expression
1		sbc2ie.1	|- A e. _V
2		sbc2ie.2	|- B e. _V
3		sbc2ie.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4		nfv	|- F/ x ps
5		nfv	|- F/ y ps
6	2	nfth	|- F/ x B e. _V
7	4 5 6 3	sbc2iegf	|- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
8	1 2 7	mp2an	|- ( [. A / x ]. [. B / y ]. ph <-> ps )