Theorem xor3 386

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		pm5.18 385	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 360	. 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:  nbbn  387  pm5.15  1009  nbi2  1012  xorass  1504  hadnot  1599  nabbi  3121  notzfausOLD  5255  nmogtmnf  28541  nmopgtmnf  29639  limsucncmpi  33788  aiffnbandciffatnotciffb  43134  axorbciffatcxorb  43135  abnotbtaxb  43145  afv2orxorb  43421  line2ylem  44732  line2xlem  44734