ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  jaddc GIF version

Theorem jaddc 760
Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
Hypotheses
Ref Expression
jaddc.1 (φ → (DECID ψ → (¬ ψθ)))
jaddc.2 (φ → (χθ))
Assertion
Ref Expression
jaddc (φ → (DECID ψ → ((ψχ) → θ)))

Proof of Theorem jaddc
StepHypRef Expression
1 jaddc.2 . . 3 (φ → (χθ))
21imim2d 48 . 2 (φ → ((ψχ) → (ψθ)))
3 jaddc.1 . . 3 (φ → (DECID ψ → (¬ ψθ)))
4 pm2.6dc 758 . . 3 (DECID ψ → ((¬ ψθ) → ((ψθ) → θ)))
53, 4sylcom 25 . 2 (φ → (DECID ψ → ((ψθ) → θ)))
62, 5syl5d 62 1 (φ → (DECID ψ → ((ψχ) → θ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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:  pm2.54dc  789
  Copyright terms: Public domain W3C validator