Theorem xor3 382

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		pm5.18 381	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 357	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 224	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206

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 207

This theorem is referenced by:  nbbn  383  pm5.15  1015  nbi2  1018  xorass  1515  hadnot  1602  nabbib  3045  nmogtmnf  30789  nmopgtmnf  31887  limsucncmpi  36446  wl-3xorbi  37474  wl-3xornot  37482  oneptri  43269  oaordnrex  43308  omnord1ex  43317  oenord1ex  43328  aiffnbandciffatnotciffb  46916  axorbciffatcxorb  46917  abnotbtaxb  46927  afv2orxorb  47240  line2ylem  48672  line2xlem  48674