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Metamath Proof Explorer

Theorem sbiev

Description: Conversion of implicit substitution to explicit substitution. Version of sbie with a disjoint variable condition, not requiring ax-13 . See sbievw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 18-Jan-2023) Remove dependence on ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023) (Proof shortened by SN, 24-Jul-2025)

	Ref	Expression
Hypotheses	sbiev.1	⊢ Ⅎ 𝑥 𝜓
	sbiev.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )

Proof

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