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

Proof

Step	Hyp	Ref	Expression
1		sbim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
2		sbft	⊢ ( Ⅎ 𝑥 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	2	imbi1d	⊢ ( Ⅎ 𝑥 𝜑 → ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) )
4	1 3	syl5bb	⊢ ( Ⅎ 𝑥 𝜑 → ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) )