Theorem xor3 381

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		pm5.18 380	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 356	. 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  382  pm5.15  1010  nbi2  1013  xorass  1508  hadnot  1595  nabbib  3034  nmogtmnf  30652  nmopgtmnf  31750  limsucncmpi  36060  wl-3xorbi  37083  wl-3xornot  37091  oneptri  42827  oaordnrex  42866  omnord1ex  42875  oenord1ex  42886  aiffnbandciffatnotciffb  46424  axorbciffatcxorb  46425  abnotbtaxb  46435  afv2orxorb  46746  line2ylem  48010  line2xlem  48012