Theorem bj-cbv3hv2 36780

Description: Version of cbv3h 2403 with two disjoint variable conditions, which does not require ax-11 2158 nor ax-13 2371. (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 2147	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-cbv3hv2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	cbv3v2 2242	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)