Theorem ralxp3f 33217

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

Hypotheses

Ref	Expression
ral3xpf.1	⊢ Ⅎ𝑦𝜑
ral3xpf.2	⊢ Ⅎ𝑧𝜑
ral3xpf.3	⊢ Ⅎ𝑤𝜑
ral3xpf.4	⊢ Ⅎ𝑥𝜓
ral3xpf.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 3075	. 2 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑))
2		ral3xpf.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	r19.23 3238	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
4		ral3xpf.3	. . . . . . . 8 ⊢ Ⅎ𝑤𝜑
5	4	r19.23 3238	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
6	5	ralbii 3097	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
7		ral3xpf.2	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
8	7	r19.23 3238	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
9	6, 8	bitri 278	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
10	9	ralbii 3097	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
11		elxpxp 33215	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩)
12	11	imbi1i 353	. . . 4 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
13	3, 10, 12	3bitr4ri 307	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
14	13	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
15		ralcom4 3162	. . 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))
16		ralcom4 3162	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
17		ralcom4 3162	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
18		ral3xpf.4	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
19		opex 5328	. . . . . . . . 9 ⊢ ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ ∈ V
20		ral3xpf.5	. . . . . . . . 9 ⊢ (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → (𝜑 ↔ 𝜓))
21	18, 19, 20	ceqsal 3447	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ 𝜓)
22	21	ralbii 3097	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
23	17, 22	bitr3i 280	. . . . . 6 ⊢ (∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
24	23	ralbii 3097	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
25	16, 24	bitr3i 280	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
26	25	ralbii 3097	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
27	15, 26	bitr3i 280	. 2 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
28	1, 14, 27	3bitri 300	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ⟨cop 4531   × cxp 5526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4888  df-opab 5099  df-xp 5534  df-rel 5535

This theorem is referenced by:  ralxp3  33218  ralxp3es  33220  frpoins3xp3g  33346