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Theorem excxor 1310
 Description: This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
excxor ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Proof of Theorem excxor
StepHypRef Expression
1 xoranor 1309 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
2 andi 765 . . 3 (((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) ↔ (((𝜑𝜓) ∧ ¬ 𝜑) ∨ ((𝜑𝜓) ∧ ¬ 𝜓)))
3 orcom 680 . . . . 5 (((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜓 ∧ ¬ 𝜑)))
4 pm3.24 660 . . . . . 6 ¬ (𝜑 ∧ ¬ 𝜑)
54biorfi 698 . . . . 5 ((𝜓 ∧ ¬ 𝜑) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜑)))
6 andir 766 . . . . 5 (((𝜑𝜓) ∧ ¬ 𝜑) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜓 ∧ ¬ 𝜑)))
73, 5, 63bitr4ri 211 . . . 4 (((𝜑𝜓) ∧ ¬ 𝜑) ↔ (𝜓 ∧ ¬ 𝜑))
8 pm5.61 741 . . . 4 (((𝜑𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
97, 8orbi12i 714 . . 3 ((((𝜑𝜓) ∧ ¬ 𝜑) ∨ ((𝜑𝜓) ∧ ¬ 𝜓)) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
101, 2, 93bitri 204 . 2 ((𝜑𝜓) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
11 orcom 680 . 2 (((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
12 ancom 262 . . 3 ((𝜓 ∧ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
1312orbi2i 712 . 2 (((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
1410, 11, 133bitri 204 1 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 102   ↔ wb 103   ∨ wo 662   ⊻ wxo 1307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-xor 1308 This theorem is referenced by:  xordc  1324  symdifxor  3231
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