Theorem xor3 382

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		pm5.18 381	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 357	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 224	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

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  1014  nbi2  1017  xorass  1512  hadnot  1599  nabbib  3043  nmogtmnf  30799  nmopgtmnf  31897  limsucncmpi  36428  wl-3xorbi  37456  wl-3xornot  37464  oneptri  43246  oaordnrex  43285  omnord1ex  43294  oenord1ex  43305  aiffnbandciffatnotciffb  46854  axorbciffatcxorb  46855  abnotbtaxb  46865  afv2orxorb  47178  line2ylem  48601  line2xlem  48603