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Theorem excxor 1617
 Description: This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
Assertion
Ref Expression
excxor ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Proof of Theorem excxor
StepHypRef Expression
1 df-xor 1613 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 xor 1000 . 2 (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
3 ancom 448 . . 3 ((𝜓 ∧ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
43orbi2i 898 . 2 (((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
51, 2, 43bitri 286 1 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 382   ∨ wo 836   ⊻ wxo 1612 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 197  df-an 383  df-or 837  df-xor 1613 This theorem is referenced by:  f1omvdco2  18075  psgnunilem5  18121  or3or  38845
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