Theorem cbv3hv 2340

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3h 2405 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2373. Was used in a proof of axc11n 2427 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)

Hypotheses

Ref	Expression
cbv3hv.nf1	⊢ (𝜑 → ∀𝑦𝜑)
cbv3hv.nf2	⊢ (𝜓 → ∀𝑥𝜓)
cbv3hv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3hv	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbv3hv

Step	Hyp	Ref	Expression
1		cbv3hv.nf1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	nf5i 2145	. 2 ⊢ Ⅎ𝑦𝜑
3		cbv3hv.nf2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2145	. 2 ⊢ Ⅎ𝑥𝜓
5		cbv3hv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	2, 4, 5	cbv3v 2335	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-11 2157  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-ex 1786  df-nf 1790

This theorem is referenced by: (None)