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Metamath Proof Explorer

Theorem cbvaldw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvaldw.1	⊢ Ⅎ 𝑦 𝜑
	cbvaldw.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	cbvaldw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	cbvaldw	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	⊢ Ⅎ 𝑦 𝜑
2		cbvaldw.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		cbvaldw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4		nfv	⊢ Ⅎ 𝑥 𝜑
5		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
6	4 1 2 5 3	cbv2w	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )