Theorem r3al

Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995) (Proof shortened by Wolf Lammen, 30-Dec-2019)

	Ref	Expression
Assertion	r3al	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		r2al	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ∈ 𝐶 𝜑 ) )
2		19.21v	⊢ ( ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐶 → 𝜑 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ( 𝑧 ∈ 𝐶 → 𝜑 ) ) )
3		df-3an	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐶 ) )
4	3	imbi1i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) ↔ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) )
5		impexp	⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐶 → 𝜑 ) ) )
6	4 5	bitri	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐶 → 𝜑 ) ) )
7	6	albii	⊢ ( ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) ↔ ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐶 → 𝜑 ) ) )
8		df-ral	⊢ ( ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝐶 → 𝜑 ) )
9	8	imbi2i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ∈ 𝐶 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ( 𝑧 ∈ 𝐶 → 𝜑 ) ) )
10	2 7 9	3bitr4ri	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ∈ 𝐶 𝜑 ) ↔ ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) )
11	10	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ∈ 𝐶 𝜑 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) )
12	1 11	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 ) )