MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xorneg Structured version   Visualization version   GIF version

Theorem xorneg 1646
Description: The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
xorneg ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))

Proof of Theorem xorneg
StepHypRef Expression
1 xorneg1 1645 . 2 ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ ¬ 𝜓))
2 xorneg2 1644 . . 3 ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32con2bii 349 . 2 ((𝜑𝜓) ↔ ¬ (𝜑 ⊻ ¬ 𝜓))
41, 3bitr4i 270 1 ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wxo 1634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-xor 1635
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator