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Theorem orbidi 898
 Description: Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.)
Assertion
Ref Expression
orbidi ((φ (ψχ)) ↔ ((φ ψ) ↔ (φ χ)))

Proof of Theorem orbidi
StepHypRef Expression
1 pm5.74 235 . 2 ((¬ φ → (ψχ)) ↔ ((¬ φψ) ↔ (¬ φχ)))
2 df-or 359 . 2 ((φ (ψχ)) ↔ (¬ φ → (ψχ)))
3 df-or 359 . . 3 ((φ ψ) ↔ (¬ φψ))
4 df-or 359 . . 3 ((φ χ) ↔ (¬ φχ))
53, 4bibi12i 306 . 2 (((φ ψ) ↔ (φ χ)) ↔ ((¬ φψ) ↔ (¬ φχ)))
61, 2, 53bitr4i 268 1 ((φ (ψχ)) ↔ ((φ ψ) ↔ (φ χ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357 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 This theorem is referenced by:  pm5.7  900
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