Theorem bj-cbval2v 31924

Description: Version of cbval2 2267 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-cbval2v.1	⊢ Ⅎ𝑧𝜑
bj-cbval2v.2	⊢ Ⅎ𝑤𝜑
bj-cbval2v.3	⊢ Ⅎ𝑥𝜓
bj-cbval2v.4	⊢ Ⅎ𝑦𝜓
bj-cbval2v.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-cbval2v	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem bj-cbval2v

Step	Hyp	Ref	Expression
1		bj-cbval2v.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfal 2139	. 2 ⊢ Ⅎ𝑧∀𝑦𝜑
3		bj-cbval2v.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2139	. 2 ⊢ Ⅎ𝑥∀𝑤𝜓
5		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧
6		bj-cbval2v.2	. . . . . 6 ⊢ Ⅎ𝑤𝜑
7	5, 6	nfim 1813	. . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 → 𝜑)
8		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧
9		bj-cbval2v.4	. . . . . 6 ⊢ Ⅎ𝑦𝜓
10	8, 9	nfim 1813	. . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑧 → 𝜓)
11		bj-cbval2v.5	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
12	11	expcom 450	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)))
13	12	pm5.74d 261	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜓)))
14	7, 10, 13	cbvalv1 2163	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑤(𝑥 = 𝑧 → 𝜓))
15		19.21v 1855	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑))
16		19.21v 1855	. . . 4 ⊢ (∀𝑤(𝑥 = 𝑧 → 𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
17	14, 15, 16	3bitr3i 289	. . 3 ⊢ ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
18	17	pm5.74ri 260	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
19	2, 4, 18	cbvalv1 2163	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀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-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-cbvex2v  31925  bj-cbval2vv  31926