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

Theorem xornbi 1349
Description: A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1354. (Contributed by Jim Kingdon, 10-Mar-2018.)
Assertion
Ref Expression
xornbi ((𝜑𝜓) → ¬ (𝜑𝜓))

Proof of Theorem xornbi
StepHypRef Expression
1 xorbin 1347 . 2 ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))
2 pm5.18im 1348 . . 3 ((𝜑𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
32con2i 601 . 2 ((𝜑 ↔ ¬ 𝜓) → ¬ (𝜑𝜓))
41, 3syl 14 1 ((𝜑𝜓) → ¬ (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wxo 1338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-xor 1339
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator