Theorem ralxp3 8117

Description: Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 2-Feb-2025.)

Hypothesis

Ref	Expression
ralxp3.1	⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralxp3	⊢ (∀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   𝜑,𝑤,𝑦,𝑧   𝜓,𝑥

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

Proof of Theorem ralxp3

Step	Hyp	Ref	Expression
1		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1914	. 2 ⊢ Ⅎ𝑧𝜑
3		nfv 1914	. 2 ⊢ Ⅎ𝑤𝜑
4		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
5		ralxp3.1	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	ralxp3f 8116	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∀wral 3044  ⟨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: (None)