Theorem wl-sbrimt

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim . (Contributed by Wolf Lammen, 26-Jul-2019)

	Ref	Expression
Assertion	wl-sbrimt	|- ( F/ x ph -> ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbft	|- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
3	2	imbi1d	|- ( F/ x ph -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) ) )
4	1 3	bitrid	|- ( F/ x ph -> ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) )