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Theorem orbididc 859
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)) ↔ ((φ ψ) ↔ (φ χ))))

Proof of Theorem orbididc
StepHypRef Expression
1 orimdidc 811 . . 3 (DECID φ → ((φ (ψχ)) ↔ ((φ ψ) → (φ χ))))
2 orimdidc 811 . . 3 (DECID φ → ((φ (χψ)) ↔ ((φ χ) → (φ ψ))))
31, 2anbi12d 442 . 2 (DECID φ → (((φ (ψχ)) (φ (χψ))) ↔ (((φ ψ) → (φ χ)) ((φ χ) → (φ ψ)))))
4 dfbi2 368 . . . 4 ((ψχ) ↔ ((ψχ) (χψ)))
54orbi2i 678 . . 3 ((φ (ψχ)) ↔ (φ ((ψχ) (χψ))))
6 ordi 728 . . 3 ((φ ((ψχ) (χψ))) ↔ ((φ (ψχ)) (φ (χψ))))
75, 6bitri 173 . 2 ((φ (ψχ)) ↔ ((φ (ψχ)) (φ (χψ))))
8 dfbi2 368 . 2 (((φ ψ) ↔ (φ χ)) ↔ (((φ ψ) → (φ χ)) ((φ χ) → (φ ψ))))
93, 7, 83bitr4g 212 1 (DECID φ → ((φ (ψχ)) ↔ ((φ ψ) ↔ (φ χ))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  pm5.7dc  860
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