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Theorem ordi 834
 Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)
Assertion
Ref Expression
ordi ((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Proof of Theorem ordi
StepHypRef Expression
1 jcab 833 . 2 ((¬ φ → (ψ χ)) ↔ ((¬ φψ) φχ)))
2 df-or 359 . 2 ((φ (ψ χ)) ↔ (¬ φ → (ψ χ)))
3 df-or 359 . . 3 ((φ ψ) ↔ (¬ φψ))
4 df-or 359 . . 3 ((φ χ) ↔ (¬ φχ))
53, 4anbi12i 678 . 2 (((φ ψ) (φ χ)) ↔ ((¬ φψ) φχ)))
61, 2, 53bitr4i 268 1 ((φ (ψ χ)) ↔ ((φ ψ) (φ 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-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:  ordir  835  orddi  839  pm5.63  890  pm4.43  893  cadan  1392  undi  3502  undif4  3607
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