MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xor3 Structured version   Visualization version   GIF version

Theorem xor3 386
Description: Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.)
Assertion
Ref Expression
xor3 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem xor3
StepHypRef Expression
1 pm5.18 385 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
21con2bii 360 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32bicomi 226 1 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208
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 209
This theorem is referenced by:  nbbn  387  pm5.15  1009  nbi2  1012  xorass  1506  hadnot  1603  nabbi  3108  notzfausOLD  5235  nmogtmnf  28528  nmopgtmnf  29626  limsucncmpi  33797  wl-3xorbi  34763  wl-3xornot  34771  aiffnbandciffatnotciffb  43284  axorbciffatcxorb  43285  abnotbtaxb  43295  afv2orxorb  43571  line2ylem  44921  line2xlem  44923
  Copyright terms: Public domain W3C validator