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	⊢ Ⅎ 𝑥 𝜑
Assertion	sbrim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbrim.1	⊢ Ⅎ 𝑥 𝜑
2		sbim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
3	1	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
4	3	imbi1i	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
5	2 4	bitri	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )