Theorem 2ralor 3109

Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)

Assertion

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

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

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

Proof of Theorem 2ralor

Step	Hyp	Ref	Expression
1		rexnal 2995	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
2		rexnal 2995	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 𝜓)
3	1, 2	anbi12i 733	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 𝜑 ∧ ¬ ∀𝑦 ∈ 𝐵 𝜓))
4		ioran 511	. . . . . . 7 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
5	4	rexbii 3041	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ (𝜑 ∨ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓))
6		rexnal 2995	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ (𝜑 ∨ 𝜓) ↔ ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
7	5, 6	bitr3i 266	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
8	7	rexbii 3041	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
9		reeanv 3107	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
10		rexnal 2995	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
11	8, 9, 10	3bitr3ri 291	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
12		ioran 511	. . 3 ⊢ (¬ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 𝜑 ∧ ¬ ∀𝑦 ∈ 𝐵 𝜓))
13	3, 11, 12	3bitr4i 292	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ¬ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
14	13	con4bii 311	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  ispridl2  33837