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 359	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 226	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208

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 209

This theorem is referenced by:  nbbnOLD  386  pm5.15  1026  nbi2  1029  xorass  1535  hadnot  1622  nabbib  3060  vn0  4297  nmogtmnf  30970  nmopgtmnf  32068  limsucncmpi  36802  wl-3xorbi  37964  wl-3xornot  37972  oneptri  43831  oaordnrex  43869  omnord1ex  43878  oenord1ex  43889  aiffnbandciffatnotciffb  47495  axorbciffatcxorb  47496  abnotbtaxb  47506  afv2orxorb  47819  line2ylem  49370  line2xlem  49372