ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm2.85dc Structured version   GIF version

Theorem pm2.85dc 799
Description: Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
pm2.85dc (DECID φ → (((φ ψ) → (φ χ)) → (φ (ψχ))))

Proof of Theorem pm2.85dc
StepHypRef Expression
1 df-dc 731 . 2 (DECID φ ↔ (φ ¬ φ))
2 orc 620 . . . 4 (φ → (φ (ψχ)))
32a1d 22 . . 3 (φ → (((φ ψ) → (φ χ)) → (φ (ψχ))))
4 olc 619 . . . . . 6 (ψ → (φ ψ))
54imim1i 54 . . . . 5 (((φ ψ) → (φ χ)) → (ψ → (φ χ)))
6 orel1 631 . . . . 5 φ → ((φ χ) → χ))
75, 6syl9r 67 . . . 4 φ → (((φ ψ) → (φ χ)) → (ψχ)))
8 olc 619 . . . 4 ((ψχ) → (φ (ψχ)))
97, 8syl6 29 . . 3 φ → (((φ ψ) → (φ χ)) → (φ (ψχ))))
103, 9jaoi 623 . 2 ((φ ¬ φ) → (((φ ψ) → (φ χ)) → (φ (ψχ))))
111, 10sylbi 114 1 (DECID φ → (((φ ψ) → (φ χ)) → (φ (ψχ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616  DECID wdc 730
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 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 731
This theorem is referenced by:  orimdidc  800
  Copyright terms: Public domain W3C validator