Theorem ralxp3 33218

Description: Restricted for-all over a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)

Hypothesis

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

Assertion

Ref	Expression
ralxp3	⊢ (∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)

Distinct variable groups:   𝐴,𝑝,𝑥,𝑦,𝑧   𝐵,𝑝,𝑥,𝑦,𝑧   𝐶,𝑝,𝑥,𝑦,𝑧   𝜓,𝑝   𝑥,𝑝,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

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

Proof of Theorem ralxp3

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1915	. 2 ⊢ Ⅎ𝑧𝜑
4		nfv 1915	. 2 ⊢ Ⅎ𝑝𝜓
5		ralxp3.1	. 2 ⊢ (𝑝 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	ralxp3f 33217	1 ⊢ (∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∀wral 3070  ⟨cop 4531   × cxp 5526

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4888  df-opab 5099  df-xp 5534  df-rel 5535

This theorem is referenced by:  no3indslem  33698