Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  con2bidc Structured version   GIF version

Theorem con2bidc 768
 Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
con2bidc (DECID φ → (DECID ψ → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))))

Proof of Theorem con2bidc
StepHypRef Expression
1 con1bidc 767 . . . . 5 (DECID φ → (DECID ψ → ((¬ φψ) ↔ (¬ ψφ))))
21imp 115 . . . 4 ((DECID φ DECID ψ) → ((¬ φψ) ↔ (¬ ψφ)))
3 bicom 128 . . . 4 ((¬ φψ) ↔ (ψ ↔ ¬ φ))
4 bicom 128 . . . 4 ((¬ ψφ) ↔ (φ ↔ ¬ ψ))
52, 3, 43bitr3g 211 . . 3 ((DECID φ DECID ψ) → ((ψ ↔ ¬ φ) ↔ (φ ↔ ¬ ψ)))
65bicomd 129 . 2 ((DECID φ DECID ψ) → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ)))
76ex 108 1 (DECID φ → (DECID ψ → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))))
 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:  annimdc  844  pm4.55dc  845  nbbndc  1282
 Copyright terms: Public domain W3C validator