Theorem xor3 385

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		pm5.18 384	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 360	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 227	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 209

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 210

This theorem is referenced by:  nbbnOLD  387  pm5.15  1030  nbi2  1033  xorass  1545  hadnot  1632  nabbib  3063  vn0OLD  4299  nmogtmnf  31122  nmopgtmnf  32220  limsucncmpi  36956  wl-3xorbi  38119  wl-3xornot  38127  oneptri  43984  oaordnrex  44022  omnord1ex  44031  oenord1ex  44042  aiffnbandciffatnotciffb  47641  axorbciffatcxorb  47642  abnotbtaxb  47652  afv2orxorb  47965  line2ylem  49531  line2xlem  49533