Theorem cbv1v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv1 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 16-Jun-2019)

	Ref	Expression
Hypotheses	cbv1v.1	⊢ Ⅎ 𝑥 𝜑
	cbv1v.2	⊢ Ⅎ 𝑦 𝜑
	cbv1v.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	cbv1v.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
	cbv1v.5	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
Assertion	cbv1v	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbv1v.1	⊢ Ⅎ 𝑥 𝜑
2		cbv1v.2	⊢ Ⅎ 𝑦 𝜑
3		cbv1v.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
4		cbv1v.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5		cbv1v.5	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
6	2 3	nfim1	⊢ Ⅎ 𝑦 ( 𝜑 → 𝜓 )
7	1 4	nfim1	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜒 )
8	5	com12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
9	8	a2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
10	6 7 9	cbv3v	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑦 ( 𝜑 → 𝜒 ) )
11	1	19.21	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )
12	2	19.21	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) )
13	10 11 12	3imtr3i	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( 𝜑 → ∀ 𝑦 𝜒 ) )
14	13	pm2.86i	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )