Theorem ralxp3f 33713

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 3060	. 2 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑))
2		ral3xpf.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	r19.23 3275	. . . 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 3275	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
6	5	ralbii 3090	. . . . . 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 3275	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 (∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
9	6, 8	bitri 274	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
10	9	ralbii 3090	. . . 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 33711	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩)
12	11	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
13	3, 10, 12	3bitr4ri 303	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
14	13	albii 1817	. 2 ⊢ (∀𝑥(𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
15		ralcom4 3258	. . 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 3258	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
17		ralcom4 3258	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑))
18		ral3xpf.4	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
19		opex 5382	. . . . . . . . 9 ⊢ ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ ∈ V
20		ral3xpf.5	. . . . . . . . 9 ⊢ (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → (𝜑 ↔ 𝜓))
21	18, 19, 20	ceqsal 3468	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ 𝜓)
22	21	ralbii 3090	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥(𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
23	17, 22	bitr3i 276	. . . . . 6 ⊢ (∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑤 ∈ 𝐶 𝜓)
24	23	ralbii 3090	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑥∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
25	16, 24	bitr3i 276	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
26	25	ralbii 3090	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → 𝜑) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
27	15, 26	bitr3i 276	. 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 296	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1535   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2101  ∀wral 3059  ∃wrex 3068  ⟨cop 4570   × cxp 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-iun 4929  df-opab 5140  df-xp 5597  df-rel 5598

This theorem is referenced by:  ralxp3  33714  ralxp3es  33716  frpoins3xp3g  33816