Theorem xor3 383

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		pm5.18 382	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 357	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 223	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205

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 206

This theorem is referenced by:  nbbn  384  pm5.15  1009  nbi2  1012  xorass  1510  hadnot  1607  nabbi  3048  nmogtmnf  29111  nmopgtmnf  30209  limsucncmpi  34613  wl-3xorbi  35623  wl-3xornot  35631  aiffnbandciffatnotciffb  44350  axorbciffatcxorb  44351  abnotbtaxb  44361  afv2orxorb  44671  line2ylem  46049  line2xlem  46051