Theorem xor3 387

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		pm5.18 386	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 361	. 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:  nbbn  388  pm5.15  1010  nbi2  1013  xorass  1507  hadnot  1604  nabbi  3089  notzfausOLD  5228  nmogtmnf  28553  nmopgtmnf  29651  limsucncmpi  33906  wl-3xorbi  34890  wl-3xornot  34898  aiffnbandciffatnotciffb  43497  axorbciffatcxorb  43498  abnotbtaxb  43508  afv2orxorb  43784  line2ylem  45165  line2xlem  45167