Theorem ralunb

Description: Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Assertion	ralunb	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elunant	⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → 𝜑 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
3		19.26	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ( 𝑥 ∈ 𝐵 → 𝜑 ) ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → 𝜑 ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
5		df-ral	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → 𝜑 ) )
6		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
7		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) )
8	6 7	anbi12i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐵 𝜑 ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
9	4 5 8	3bitr4i	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐵 𝜑 ) )