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Theorem xorbin 1362
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 1354 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 imnan 679 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32biimpri 132 . . . 4 (¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
43adantl 275 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (𝜑 → ¬ 𝜓))
51, 4sylbi 120 . 2 ((𝜑𝜓) → (𝜑 → ¬ 𝜓))
6 pm2.53 711 . . . . 5 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 719 . . . 4 ((𝜑𝜓) → (¬ 𝜓𝜑))
87adantr 274 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (¬ 𝜓𝜑))
91, 8sylbi 120 . 2 ((𝜑𝜓) → (¬ 𝜓𝜑))
105, 9impbid 128 1 ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  wxo 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-xor 1354
This theorem is referenced by:  xornbi  1364  zeo4  11578  odd2np1  11581
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