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))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  nbbnOLD  387  pm5.15  1030  nbi2  1033  xorass  1545  hadnot  1632  had0  1634  nabbib  3065  vn0OLD  4299  nmogtmnf  31193  nmopgtmnf  32291  limsucncmpi  37013  wl-3xorbi  38176  wl-3xornot  38184  oneptri  44042  oaordnrex  44080  omnord1ex  44089  oenord1ex  44100  aiffnbandciffatnotciffb  47699  axorbciffatcxorb  47700  abnotbtaxb  47710  afv2orxorb  48023  line2ylem  49588  line2xlem  49590