Theorem ralxp3f 8184

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 3081	. 2 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑))
2		el2xptp 5827	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩)
3	2	imbi1i 352	. . . 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 3263	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
6	5	ralbii 3112	. . . . . . 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 3263	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
9	6, 8	bitri 278	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
10	9	ralbii 3112	. . . . 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 3263	. . . . 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 279	. . . 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 278	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
15	14	albii 1852	. 2 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
16		ralcom4 3292	. . 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 3292	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
18		ralcom4 3292	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
19		ralxp3f.4	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
20		otex 5439	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑧, 𝑤⟩ ∈ V
21		ralxp3f.5	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
22	19, 20, 21	ceqsal 3491	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ 𝜓)
23	22	ralbii 3112	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
24	18, 23	bitr3i 280	. . . . . 6 ⊢ (∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
25	24	ralbii 3112	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
26	17, 25	bitr3i 280	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
27	26	ralbii 3112	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
28	16, 27	bitr3i 280	. 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 300	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⟨cotp 4861   × cxp 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-iun 4956  df-opab 5172  df-xp 5662  df-rel 5663

This theorem is used by:  ralxp3  8185  ralxp3es  8186  frpoins3xp3g  8188