Theorem xor3 384

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		pm5.18 383	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 358	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 223	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205

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 206

This theorem is referenced by:  nbbn  385  pm5.15  1012  nbi2  1015  xorass  1515  hadnot  1604  nabbib  3046  nmogtmnf  30023  nmopgtmnf  31121  limsucncmpi  35330  wl-3xorbi  36354  wl-3xornot  36362  oneptri  42006  oaordnrex  42045  omnord1ex  42054  oenord1ex  42065  aiffnbandciffatnotciffb  45614  axorbciffatcxorb  45615  abnotbtaxb  45625  afv2orxorb  45936  line2ylem  47437  line2xlem  47439