Theorem 2ralor 2780

Description: Distribute 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 2625	. . . 4 ⊢ (∃x ∈ A ¬ φ ↔ ¬ ∀x ∈ A φ)
2		rexnal 2625	. . . 4 ⊢ (∃y ∈ B ¬ ψ ↔ ¬ ∀y ∈ B ψ)
3	1, 2	anbi12i 678	. . 3 ⊢ ((∃x ∈ A ¬ φ ∧ ∃y ∈ B ¬ ψ) ↔ (¬ ∀x ∈ A φ ∧ ¬ ∀y ∈ B ψ))
4		ioran 476	. . . . . . 7 ⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))
5	4	rexbii 2639	. . . . . 6 ⊢ (∃y ∈ B ¬ (φ ∨ ψ) ↔ ∃y ∈ B (¬ φ ∧ ¬ ψ))
6		rexnal 2625	. . . . . 6 ⊢ (∃y ∈ B ¬ (φ ∨ ψ) ↔ ¬ ∀y ∈ B (φ ∨ ψ))
7	5, 6	bitr3i 242	. . . . 5 ⊢ (∃y ∈ B (¬ φ ∧ ¬ ψ) ↔ ¬ ∀y ∈ B (φ ∨ ψ))
8	7	rexbii 2639	. . . 4 ⊢ (∃x ∈ A ∃y ∈ B (¬ φ ∧ ¬ ψ) ↔ ∃x ∈ A ¬ ∀y ∈ B (φ ∨ ψ))
9		reeanv 2778	. . . 4 ⊢ (∃x ∈ A ∃y ∈ B (¬ φ ∧ ¬ ψ) ↔ (∃x ∈ A ¬ φ ∧ ∃y ∈ B ¬ ψ))
10		rexnal 2625	. . . 4 ⊢ (∃x ∈ A ¬ ∀y ∈ B (φ ∨ ψ) ↔ ¬ ∀x ∈ A ∀y ∈ B (φ ∨ ψ))
11	8, 9, 10	3bitr3ri 267	. . 3 ⊢ (¬ ∀x ∈ A ∀y ∈ B (φ ∨ ψ) ↔ (∃x ∈ A ¬ φ ∧ ∃y ∈ B ¬ ψ))
12		ioran 476	. . 3 ⊢ (¬ (∀x ∈ A φ ∨ ∀y ∈ B ψ) ↔ (¬ ∀x ∈ A φ ∧ ¬ ∀y ∈ B ψ))
13	3, 11, 12	3bitr4i 268	. 2 ⊢ (¬ ∀x ∈ A ∀y ∈ B (φ ∨ ψ) ↔ ¬ (∀x ∈ A φ ∨ ∀y ∈ B ψ))
14	13	con4bii 288	1 ⊢ (∀x ∈ A ∀y ∈ B (φ ∨ ψ) ↔ (∀x ∈ A φ ∨ ∀y ∈ B ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wral 2614  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620

This theorem is referenced by: (None)