Theorem xor3 383

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		pm5.18 382	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 358	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 225	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 207

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 208

This theorem is referenced by:  nbbn  384  pm5.15  1020  nbi2  1023  xorass  1522  hadnot  1609  nabbib  3037  vn0  4273  nmogtmnf  30859  nmopgtmnf  31957  limsucncmpi  36673  wl-3xorbi  37835  wl-3xornot  37843  oneptri  43702  oaordnrex  43740  omnord1ex  43749  oenord1ex  43760  aiffnbandciffatnotciffb  47367  axorbciffatcxorb  47368  abnotbtaxb  47378  afv2orxorb  47691  line2ylem  49242  line2xlem  49244