Theorem cbv3v 2343

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 2405 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbv3v.nf1	⊢ Ⅎ𝑦𝜑
cbv3v.nf2	⊢ Ⅎ𝑥𝜓
cbv3v.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3v	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbv3v

Step	Hyp	Ref	Expression
1		cbv3v.nf1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nf5ri 2207	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbal 2178	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
4		cbv3v.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
5		cbv3v.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	4, 5	spimfv 2251	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
7	3, 6	alrimih 1831	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbv1v  2344  cbv3hv  2348  cbvalv1  2349