Theorem ralxp3 8140

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 1947	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1947	. 2 ⊢ Ⅎ𝑧𝜑
3		nfv 1947	. 2 ⊢ Ⅎ𝑤𝜑
4		nfv 1947	. 2 ⊢ Ⅎ𝑥𝜓
5		ralxp3.1	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	ralxp3f 8139	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∀wral 3081  ⟨cotp 4599   × cxp 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-iun 4960  df-opab 5176  df-xp 5669  df-rel 5670

This theorem is used by: (None)