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  1515  hadnot  1602  nabbib  3035  nmogtmnf  30751  nmopgtmnf  31849  limsucncmpi  36463  wl-3xorbi  37491  wl-3xornot  37499  oneptri  43281  oaordnrex  43319  omnord1ex  43328  oenord1ex  43339  aiffnbandciffatnotciffb  46933  axorbciffatcxorb  46934  abnotbtaxb  46944  afv2orxorb  47257  line2ylem  48731  line2xlem  48733