Theorem cbval2vOLD 2343

Description: Obsolete version of cbval2v 2342 as of 14-Jan-2024. (Contributed by BJ, 16-Jan-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbval2v.1	⊢ Ⅎ𝑧𝜑
cbval2v.2	⊢ Ⅎ𝑤𝜑
cbval2v.3	⊢ Ⅎ𝑥𝜓
cbval2v.4	⊢ Ⅎ𝑦𝜓
cbval2v.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbval2vOLD	⊢ (∀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 cbval2vOLD

Step	Hyp	Ref	Expression
1		cbval2v.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfal 2321	. 2 ⊢ Ⅎ𝑧∀𝑦𝜑
3		cbval2v.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2321	. 2 ⊢ Ⅎ𝑥∀𝑤𝜓
5		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧
6		cbval2v.2	. . . . . 6 ⊢ Ⅎ𝑤𝜑
7	5, 6	nfim 1900	. . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 → 𝜑)
8		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧
9		cbval2v.4	. . . . . 6 ⊢ Ⅎ𝑦𝜓
10	8, 9	nfim 1900	. . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑧 → 𝜓)
11		cbval2v.5	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
12	11	expcom 413	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)))
13	12	pm5.74d 272	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜓)))
14	7, 10, 13	cbvalv1 2340	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑤(𝑥 = 𝑧 → 𝜓))
15		19.21v 1943	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑))
16		19.21v 1943	. . . 4 ⊢ (∀𝑤(𝑥 = 𝑧 → 𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
17	14, 15, 16	3bitr3i 300	. . 3 ⊢ ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
18	17	pm5.74ri 271	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
19	2, 4, 18	cbvalv1 2340	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)