Theorem ralxp3f 8116

Description: Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
ralxp3f.1	⊢ Ⅎ𝑦𝜑
ralxp3f.2	⊢ Ⅎ𝑧𝜑
ralxp3f.3	⊢ Ⅎ𝑤𝜑
ralxp3f.4	⊢ Ⅎ𝑥𝜓
ralxp3f.5	⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralxp3f	⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Distinct variable groups:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑥,𝑦,𝑧

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

Proof of Theorem ralxp3f

Step	Hyp	Ref	Expression
1		df-ral 3045	. 2 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑))
2		el2xptp 8014	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩)
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
4		ralxp3f.3	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
5	4	r19.23 3234	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
6	5	ralbii 3075	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
7		ralxp3f.2	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
8	7	r19.23 3234	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
9	6, 8	bitri 275	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
10	9	ralbii 3075	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
11		ralxp3f.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
12	11	r19.23 3234	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
13	10, 12	bitr2i 276	. . . 4 ⊢ ((∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
14	3, 13	bitri 275	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
15	14	albii 1819	. 2 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
16		ralcom4 3263	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
17		ralcom4 3263	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
18		ralcom4 3263	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
19		ralxp3f.4	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
20		otex 5425	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑧, 𝑤⟩ ∈ V
21		ralxp3f.5	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
22	19, 20, 21	ceqsal 3485	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ 𝜓)
23	22	ralbii 3075	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
24	18, 23	bitr3i 277	. . . . . 6 ⊢ (∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
25	24	ralbii 3075	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
26	17, 25	bitr3i 277	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
27	26	ralbii 3075	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
28	16, 27	bitr3i 277	. 2 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
29	1, 15, 28	3bitri 297	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ⟨cotp 4597   × cxp 5636

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-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-iun 4957  df-opab 5170  df-xp 5644  df-rel 5645

This theorem is referenced by:  ralxp3  8117  ralxp3es  8118  frpoins3xp3g  8120