Theorem sbc19.21g

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004)

	Ref	Expression
Hypothesis	sbcgf.1	|- F/ x ph
Assertion	sbc19.21g	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbcgf.1	|- F/ x ph
2		sbcimg	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
3	1	sbcgf	|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
4	3	imbi1d	|- ( A e. V -> ( ( [. A / x ]. ph -> [. A / x ]. ps ) <-> ( ph -> [. A / x ]. ps ) ) )
5	2 4	bitrd	|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )