Theorem sbrimv

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim not depending on ax-10 , but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024)

	Ref	Expression
Hypothesis	sbrim.1	⊢ Ⅎ 𝑥 𝜑
Assertion	sbrimv	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbrim.1	⊢ Ⅎ 𝑥 𝜑
2	1	19.21	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) )
3	2	sbrimvlem	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )