Theorem dn1 932

Description: A single axiom for Boolean algebra known as DN₁. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jeffrey Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Assertion

Ref	Expression
dn1	⊢ (¬ (¬ (¬ (φ ∨ ψ) ∨ χ) ∨ ¬ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))) ↔ χ)

Proof of Theorem dn1

Step	Hyp	Ref	Expression
1		pm2.45 386	. . . . 5 ⊢ (¬ (φ ∨ ψ) → ¬ φ)
2		imnan 411	. . . . 5 ⊢ ((¬ (φ ∨ ψ) → ¬ φ) ↔ ¬ (¬ (φ ∨ ψ) ∧ φ))
3	1, 2	mpbi 199	. . . 4 ⊢ ¬ (¬ (φ ∨ ψ) ∧ φ)
4	3	biorfi 396	. . 3 ⊢ (χ ↔ (χ ∨ (¬ (φ ∨ ψ) ∧ φ)))
5		orcom 376	. . . 4 ⊢ ((χ ∨ (¬ (φ ∨ ψ) ∧ φ)) ↔ ((¬ (φ ∨ ψ) ∧ φ) ∨ χ))
6		ordir 835	. . . 4 ⊢ (((¬ (φ ∨ ψ) ∧ φ) ∨ χ) ↔ ((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ χ)))
7	5, 6	bitri 240	. . 3 ⊢ ((χ ∨ (¬ (φ ∨ ψ) ∧ φ)) ↔ ((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ χ)))
8	4, 7	bitri 240	. 2 ⊢ (χ ↔ ((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ χ)))
9		pm4.45 669	. . . . 5 ⊢ (χ ↔ (χ ∧ (χ ∨ θ)))
10		anor 475	. . . . 5 ⊢ ((χ ∧ (χ ∨ θ)) ↔ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))
11	9, 10	bitri 240	. . . 4 ⊢ (χ ↔ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))
12	11	orbi2i 505	. . 3 ⊢ ((φ ∨ χ) ↔ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ))))
13	12	anbi2i 675	. 2 ⊢ (((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ χ)) ↔ ((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))))
14		anor 475	. 2 ⊢ (((¬ (φ ∨ ψ) ∨ χ) ∧ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))) ↔ ¬ (¬ (¬ (φ ∨ ψ) ∨ χ) ∨ ¬ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))))
15	8, 13, 14	3bitrri 263	1 ⊢ (¬ (¬ (¬ (φ ∨ ψ) ∨ χ) ∨ ¬ (φ ∨ ¬ (¬ χ ∨ ¬ (χ ∨ θ)))) ↔ χ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by: (None)