Theorem bj-cbv3hv2 37489

Description: Version of cbv3h 2438 with two disjoint variable conditions, which does not require ax-11 2195 nor ax-13 2406. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv3hv2.nf	⊢ (𝜓 → ∀𝑥𝜓)
bj-cbv3hv2.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-cbv3hv2	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem bj-cbv3hv2

Step	Hyp	Ref	Expression
1		bj-cbv3hv2.nf	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2184	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-cbv3hv2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	cbv3v2 2280	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

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

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-10 2179  ax-12 2216

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

This theorem is used by: (None)