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

Theorem nbbndc 1282
 Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
nbbndc (DECID φ → (DECID ψ → ((¬ φψ) ↔ ¬ (φψ))))

Proof of Theorem nbbndc
StepHypRef Expression
1 xor3dc 1275 . . . . 5 (DECID φ → (DECID ψ → (¬ (φψ) ↔ (φ ↔ ¬ ψ))))
21imp 115 . . . 4 ((DECID φ DECID ψ) → (¬ (φψ) ↔ (φ ↔ ¬ ψ)))
3 con2bidc 768 . . . . 5 (DECID φ → (DECID ψ → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))))
43imp 115 . . . 4 ((DECID φ DECID ψ) → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ)))
52, 4bitrd 177 . . 3 ((DECID φ DECID ψ) → (¬ (φψ) ↔ (ψ ↔ ¬ φ)))
6 bicom 128 . . 3 ((ψ ↔ ¬ φ) ↔ (¬ φψ))
75, 6syl6rbb 186 . 2 ((DECID φ DECID ψ) → ((¬ φψ) ↔ ¬ (φψ)))
87ex 108 1 (DECID φ → (DECID ψ → ((¬ φψ) ↔ ¬ (φψ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  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:  biassdc  1283
 Copyright terms: Public domain W3C validator