Theorem sbcimg

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004)

	Ref	Expression
Assertion	sbcimg	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	|- ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2		dfsbcq2	|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2	|- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2 3	imbi12d	|- ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5		sbim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6	1 4 5	vtoclbg	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )