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Theorem xorbin 1316
Description: A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
Assertion
Ref Expression
xorbin ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))

Proof of Theorem xorbin
StepHypRef Expression
1 df-xor 1308 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 imnan 657 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32biimpri 131 . . . 4 (¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
43adantl 271 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (𝜑 → ¬ 𝜓))
51, 4sylbi 119 . 2 ((𝜑𝜓) → (𝜑 → ¬ 𝜓))
6 pm2.53 674 . . . . 5 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 682 . . . 4 ((𝜑𝜓) → (¬ 𝜓𝜑))
87adantr 270 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (¬ 𝜓𝜑))
91, 8sylbi 119 . 2 ((𝜑𝜓) → (¬ 𝜓𝜑))
105, 9impbid 127 1 ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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:  xornbi  1318  zeo4  10649  odd2np1  10652
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