ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm4.79dc Structured version   GIF version

Theorem pm4.79dc 808
Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
pm4.79dc (DECID φ → (DECID ψ → (((ψφ) (χφ)) ↔ ((ψ χ) → φ))))

Proof of Theorem pm4.79dc
StepHypRef Expression
1 id 19 . . . 4 ((ψφ) → (ψφ))
2 id 19 . . . 4 ((χφ) → (χφ))
31, 2jaoa 639 . . 3 (((ψφ) (χφ)) → ((ψ χ) → φ))
4 simplimdc 756 . . . . . 6 (DECID ψ → (¬ (ψφ) → ψ))
5 pm3.3 248 . . . . . 6 (((ψ χ) → φ) → (ψ → (χφ)))
64, 5syl9 66 . . . . 5 (DECID ψ → (((ψ χ) → φ) → (¬ (ψφ) → (χφ))))
7 dcim 783 . . . . . 6 (DECID ψ → (DECID φDECID (ψφ)))
8 pm2.54dc 789 . . . . . 6 (DECID (ψφ) → ((¬ (ψφ) → (χφ)) → ((ψφ) (χφ))))
97, 8syl6 29 . . . . 5 (DECID ψ → (DECID φ → ((¬ (ψφ) → (χφ)) → ((ψφ) (χφ)))))
106, 9syl5d 62 . . . 4 (DECID ψ → (DECID φ → (((ψ χ) → φ) → ((ψφ) (χφ)))))
1110imp 115 . . 3 ((DECID ψ DECID φ) → (((ψ χ) → φ) → ((ψφ) (χφ))))
123, 11impbid2 131 . 2 ((DECID ψ DECID φ) → (((ψφ) (χφ)) ↔ ((ψ χ) → φ)))
1312expcom 109 1 (DECID φ → (DECID ψ → (((ψφ) (χφ)) ↔ ((ψ χ) → φ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  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-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator