ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.12dc Structured version   GIF version

Theorem pm5.12dc 815
Description: Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.12dc (DECID ψ → ((φψ) (φ → ¬ ψ)))

Proof of Theorem pm5.12dc
StepHypRef Expression
1 df-dc 742 . 2 (DECID ψ ↔ (ψ ¬ ψ))
2 ax-1 5 . . 3 (ψ → (φψ))
3 ax-1 5 . . 3 ψ → (φ → ¬ ψ))
42, 3orim12i 675 . 2 ((ψ ¬ ψ) → ((φψ) (φ → ¬ ψ)))
51, 4sylbi 114 1 (DECID ψ → ((φψ) (φ → ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   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-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