Theorem sbc2ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Revised by Mario Carneiro, 19-Dec-2013) (Proof shortened by Gino Giotto, 12-Oct-2024)

	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	2	a1i	|- ( x = A -> B e. _V )
5	4 3	sbcied	|- ( x = A -> ( [. B / y ]. ph <-> ps ) )
6	1 5	sbcie	|- ( [. A / x ]. [. B / y ]. ph <-> ps )