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Theorem orbididc 860
 Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on 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 Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
orbididc (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))

Proof of Theorem orbididc
StepHypRef Expression
1 orimdidc 812 . . 3 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
2 orimdidc 812 . . 3 (DECID 𝜑 → ((𝜑 ∨ (𝜒𝜓)) ↔ ((𝜑𝜒) → (𝜑𝜓))))
31, 2anbi12d 442 . 2 (DECID 𝜑 → (((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))) ↔ (((𝜑𝜓) → (𝜑𝜒)) ∧ ((𝜑𝜒) → (𝜑𝜓)))))
4 dfbi2 368 . . . 4 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
54orbi2i 679 . . 3 ((𝜑 ∨ (𝜓𝜒)) ↔ (𝜑 ∨ ((𝜓𝜒) ∧ (𝜒𝜓))))
6 ordi 729 . . 3 ((𝜑 ∨ ((𝜓𝜒) ∧ (𝜒𝜓))) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))))
75, 6bitri 173 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))))
8 dfbi2 368 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ (((𝜑𝜓) → (𝜑𝜒)) ∧ ((𝜑𝜒) → (𝜑𝜓))))
93, 7, 83bitr4g 212 1 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  pm5.7dc  861
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