Theorem 2ralor

Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010) (Proof shortened by Wolf Lammen, 20-Nov-2024)

	Ref	Expression
Assertion	2ralor	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.32v	⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
2		orcom	⊢ ( ( 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) )
3	1 2	bitri	⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) )
5		r19.32v	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ ∀ 𝑥 ∈ 𝐴 𝜑 ) )
6		orcom	⊢ ( ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
7	4 5 6	3bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) )