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Metamath Proof Explorer

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) (Proof shortened by SN, 24-Aug-2025)

		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	1	sbbiiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜓 )
3		sbv	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜓 )
4	2 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )