Theorem ralxp3

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

	Ref	Expression
Hypothesis	ralxp3.1	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 , 𝑤 ⟩ → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralxp3	⊢ ( ∀ 𝑥 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐶 𝜓 )

Proof

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