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Theorem orbididc 897
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 848 . . 3 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
2 orimdidc 848 . . 3 (DECID 𝜑 → ((𝜑 ∨ (𝜒𝜓)) ↔ ((𝜑𝜒) → (𝜑𝜓))))
31, 2anbi12d 457 . 2 (DECID 𝜑 → (((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))) ↔ (((𝜑𝜓) → (𝜑𝜒)) ∧ ((𝜑𝜒) → (𝜑𝜓)))))
4 dfbi2 380 . . . 4 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
54orbi2i 712 . . 3 ((𝜑 ∨ (𝜓𝜒)) ↔ (𝜑 ∨ ((𝜓𝜒) ∧ (𝜒𝜓))))
6 ordi 763 . . 3 ((𝜑 ∨ ((𝜓𝜒) ∧ (𝜒𝜓))) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))))
75, 6bitri 182 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑 ∨ (𝜒𝜓))))
8 dfbi2 380 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ (((𝜑𝜓) → (𝜑𝜒)) ∧ ((𝜑𝜒) → (𝜑𝜓))))
93, 7, 83bitr4g 221 1 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by:  pm5.7dc  898
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