Theorem xor3 385

Description: Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.)

Assertion

Ref	Expression
xor3	⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem xor3

Step	Hyp	Ref	Expression
1		pm5.18 384	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 360	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 227	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 209

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 210

This theorem is referenced by:  nbbnOLD  387  pm5.15  1028  nbi2  1031  xorass  1542  hadnot  1629  nabbib  3069  vn0OLD  4307  nmogtmnf  31062  nmopgtmnf  32160  limsucncmpi  36844  wl-3xorbi  38006  wl-3xornot  38014  oneptri  43875  oaordnrex  43913  omnord1ex  43922  oenord1ex  43933  aiffnbandciffatnotciffb  47529  axorbciffatcxorb  47530  abnotbtaxb  47540  afv2orxorb  47853  line2ylem  49415  line2xlem  49417