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

Theorem xor3 382
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 pm5.18 381 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
21con2bii 357 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32bicomi 224 1 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206
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 207
This theorem is referenced by:  nbbn  383  pm5.15  1015  nbi2  1018  xorass  1515  hadnot  1602  nabbib  3045  nmogtmnf  30789  nmopgtmnf  31887  limsucncmpi  36446  wl-3xorbi  37474  wl-3xornot  37482  oneptri  43269  oaordnrex  43308  omnord1ex  43317  oenord1ex  43328  aiffnbandciffatnotciffb  46916  axorbciffatcxorb  46917  abnotbtaxb  46927  afv2orxorb  47240  line2ylem  48672  line2xlem  48674
  Copyright terms: Public domain W3C validator