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Theorem excxor 1283
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 1282 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
2 andi 740 . . 3 (((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) ↔ (((𝜑𝜓) ∧ ¬ 𝜑) ∨ ((𝜑𝜓) ∧ ¬ 𝜓)))
3 orcom 655 . . . . 5 (((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜓 ∧ ¬ 𝜑)))
4 pm3.24 635 . . . . . 6 ¬ (𝜑 ∧ ¬ 𝜑)
54biorfi 673 . . . . 5 ((𝜓 ∧ ¬ 𝜑) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜑)))
6 andir 741 . . . . 5 (((𝜑𝜓) ∧ ¬ 𝜑) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜓 ∧ ¬ 𝜑)))
73, 5, 63bitr4ri 206 . . . 4 (((𝜑𝜓) ∧ ¬ 𝜑) ↔ (𝜓 ∧ ¬ 𝜑))
8 pm5.61 716 . . . 4 (((𝜑𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
97, 8orbi12i 689 . . 3 ((((𝜑𝜓) ∧ ¬ 𝜑) ∨ ((𝜑𝜓) ∧ ¬ 𝜓)) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
101, 2, 93bitri 199 . 2 ((𝜑𝜓) ↔ ((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
11 orcom 655 . 2 (((𝜓 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
12 ancom 257 . . 3 ((𝜓 ∧ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
1312orbi2i 687 . 2 (((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
1410, 11, 133bitri 199 1 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102  wo 637  wxo 1280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638
This theorem depends on definitions:  df-bi 114  df-xor 1281
This theorem is referenced by:  xordc  1297  symdifxor  3228
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