Theorem sbievw

Description: Conversion of implicit substitution to explicit substitution. Version of sbie and sbiev with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by BJ, 18-Jul-2023)

	Ref	Expression
Hypothesis	sbievw.is	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbievw	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbievw.is	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
3	1	equsalvw	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )
4	2 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )