Theorem 2ralor 3372

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 3241	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
2		rexnal 3241	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 𝜓)
3	1, 2	anbi12i 628	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 𝜑 ∧ ¬ ∀𝑦 ∈ 𝐵 𝜓))
4		ioran 980	. . . . . . 7 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
5	4	rexbii 3250	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ (𝜑 ∨ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓))
6		rexnal 3241	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ (𝜑 ∨ 𝜓) ↔ ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
7	5, 6	bitr3i 279	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
8	7	rexbii 3250	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
9		reeanv 3370	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (¬ 𝜑 ∧ ¬ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
10		rexnal 3241	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓))
11	8, 9, 10	3bitr3ri 304	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
12		ioran 980	. . 3 ⊢ (¬ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 𝜑 ∧ ¬ ∀𝑦 ∈ 𝐵 𝜓))
13	3, 11, 12	3bitr4i 305	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ¬ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
14	13	con4bii 323	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wral 3141  ∃wrex 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-ral 3146  df-rex 3147

This theorem is referenced by:  prmidl2  30962  ispridl2  35320