Theorem sblim

Description: Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013) (Revised by Mario Carneiro, 4-Oct-2016)

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

Proof

Step	Hyp	Ref	Expression
1		sblim.1	|- F/ x ps
2		sbim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
3	1	sbf	|- ( [ y / x ] ps <-> ps )
4	3	imbi2i	|- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	2 4	bitri	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )