Theorem sbrim

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. See sbrimv for a version with disjoint variables not requiring ax-10 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016)

	Ref	Expression
Hypothesis	sbrim.1	|- F/ x ph
Assertion	sbrim	|- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Proof

Step	Hyp	Ref	Expression
1		sbrim.1	|- F/ x ph
2		sbim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
3	1	sbf	|- ( [ y / x ] ph <-> ph )
4	3	imbi1i	|- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
5	2 4	bitri	|- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )