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Theorem xor 931
 Description: Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)
Assertion
Ref Expression
xor (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Proof of Theorem xor
StepHypRef Expression
1 iman 439 . . . 4 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
2 iman 439 . . . 4 ((𝜓𝜑) ↔ ¬ (𝜓 ∧ ¬ 𝜑))
31, 2anbi12i 729 . . 3 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑)))
4 dfbi2 658 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
5 ioran 510 . . 3 (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑)))
63, 4, 53bitr4ri 292 . 2 (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (𝜑𝜓))
76con1bii 345 1 (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383 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 196  df-or 384  df-an 385 This theorem is referenced by:  dfbi3  933  pm5.24  934  4exmid  977  excxor  1461  elsymdif  3811  symdif2  3814  rpnnen2lem12  14793  ist0-3  20959  eliuniincex  38323  eliincex  38324  abnotataxb  39732  ldepslinc  42092
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