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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
StepHypRef Expression
1 pm2.45 386 . . . . 5 (¬ (φ ψ) → ¬ φ)
2 imnan 411 . . . . 5 ((¬ (φ ψ) → ¬ φ) ↔ ¬ (¬ (φ ψ) φ))
31, 2mpbi 199 . . . 4 ¬ (¬ (φ ψ) φ)
43biorfi 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)))
75, 6bitri 240 . . 3 ((χ (¬ (φ ψ) φ)) ↔ ((¬ (φ ψ) χ) (φ χ)))
84, 7bitri 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)))
119, 10bitri 240 . . . 4 (χ ↔ ¬ (¬ χ ¬ (χ θ)))
1211orbi2i 505 . . 3 ((φ χ) ↔ (φ ¬ (¬ χ ¬ (χ θ))))
1312anbi2i 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)))))
158, 13, 143bitrri 263 1 (¬ (¬ (¬ (φ ψ) χ) ¬ (φ ¬ (¬ χ ¬ (χ θ)))) ↔ χ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by: (None)
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