Theorem ralxp3f 8112

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 3076	. 2 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑))
2		el2xptp 8012	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩)
3	2	imbi1i 351	. . . 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 3258	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
6	5	ralbii 3107	. . . . . . 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 3258	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
9	6, 8	bitri 277	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
10	9	ralbii 3107	. . . . 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 3258	. . . . 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 278	. . . 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 277	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
15	14	albii 1838	. 2 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
16		ralcom4 3287	. . 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 3287	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
18		ralcom4 3287	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑))
19		ralxp3f.4	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
20		otex 5432	. . . . . . . . 9 ⊢ ⟨𝑦, 𝑧, 𝑤⟩ ∈ V
21		ralxp3f.5	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
22	19, 20, 21	ceqsal 3490	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ 𝜓)
23	22	ralbii 3107	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
24	18, 23	bitr3i 279	. . . . . 6 ⊢ (∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
25	24	ralbii 3107	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
26	17, 25	bitr3i 279	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
27	26	ralbii 3107	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
28	16, 27	bitr3i 279	. 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 299	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ⟨cotp 4589   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-ot 4590  df-iun 4950  df-opab 5162  df-xp 5651  df-rel 5652

This theorem is referenced by:  ralxp3  8113  ralxp3es  8114  frpoins3xp3g  8116