Theorem bj-cbvaldv 31922

Description: Version of cbvald 2265 with a dv condition, which does not require ax-13 2234. (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 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-cbvaldv.1	. 2 ⊢ Ⅎ𝑦𝜑
3		bj-cbvaldv.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfv 1830	. . 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 31919	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-cbvexdv  31923  bj-cbvaldvav  31928