Theorem cbv3v2 2243

Description: Version of cbv3 2415 with two disjoint variable conditions, which does not require ax-11 2161 nor ax-13 2390. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.)

Hypotheses

Ref	Expression
cbv3v2.nf	⊢ Ⅎ𝑥𝜓
cbv3v2.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3v2	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem cbv3v2

Step	Hyp	Ref	Expression
1		cbv3v2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		cbv3v2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimfv 2241	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
4	3	alrimiv 1928	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-cbv3hv2  34117