Theorem cbv3v2 2280

Description: Version of cbv3 2432 with two disjoint variable conditions, which does not require ax-11 2195 nor ax-13 2407. (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 2278	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
4	3	alrimiv 1960	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  bj-cbv3hv2  37471