Theorem 2ralor 3245

Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 20-Nov-2024.)

Assertion

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

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem 2ralor

Step	Hyp	Ref	Expression
1		r19.32v 3204	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
2		orcom 883	. . . 4 ⊢ ((𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑))
3	1, 2	bitri 278	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑))
4	3	ralbii 3117	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑))
5		r19.32v 3204	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
6		orcom 883	. 2 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
7	4, 5, 6	3bitri 300	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   ↔ wb 209   ∨ wo 860  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ral 3086

This theorem is referenced by:  prmidl2  21433  ispridl2  38572