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  1544  hadnot  1631  had0  1633  nabbib  3060  vn0OLD  4291  nmogtmnf  31379  nmopgtmnf  32477  dfac16  35812  maxssj0  35947  limsucncmpi  37327  wl-3xorbi  38503  wl-3xornot  38511  oneptri  44366  oaordnrex  44404  omnord1ex  44413  oenord1ex  44424  aiffnbandciffatnotciffb  48081  axorbciffatcxorb  48082  abnotbtaxb  48092  afv2orxorb  48405  line2ylem  49970  line2xlem  49972