Theorem cbv3v 2158

Description: Version of cbv3 2253 with a dv condition, which does not require ax-13 2234. (Contributed 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	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfal 2139	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
3		cbv3v.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
4		cbv3v.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	spimv1 2101	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
6	2, 5	alrimi 2069	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  cbv3hv  2160  cbvalv1  2163  bj-cbv1v  31916