Theorem bj-cbvaldv 34121

Description: Version of cbvald 2428 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbvaldv.1	⊢ Ⅎ𝑦𝜑
bj-cbvaldv.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
bj-cbvaldv.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
bj-cbvaldv	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem bj-cbvaldv

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-cbvaldv.1	. 2 ⊢ Ⅎ𝑦𝜑
3		bj-cbvaldv.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfv 1915	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	a1i 11	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6		bj-cbvaldv.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
7	1, 2, 3, 5, 6	bj-cbv2v 34120	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀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-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-cbvexdv  34122  bj-cbvaldvav  34125