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Metamath Proof Explorer

Theorem cbv3v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbv3v.nf1	⊢ Ⅎ 𝑦 𝜑
	cbv3v.nf2	⊢ Ⅎ 𝑥 𝜓
	cbv3v.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	cbv3v	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbv3v.nf1	⊢ Ⅎ 𝑦 𝜑
2		cbv3v.nf2	⊢ Ⅎ 𝑥 𝜓
3		cbv3v.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
5	4	hbal	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
6	2 3	spimfv	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
7	5 6	alrimih	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )