Theorem ralxp3 8088

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 1916	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1916	. 2 ⊢ Ⅎ𝑧𝜑
3		nfv 1916	. 2 ⊢ Ⅎ𝑤𝜑
4		nfv 1916	. 2 ⊢ Ⅎ𝑥𝜓
5		ralxp3.1	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	ralxp3f 8087	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∀wral 3051  ⟨cotp 4575   × cxp 5629

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-ot 4576  df-iun 4935  df-opab 5148  df-xp 5637  df-rel 5638

This theorem is referenced by: (None)