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Theorem xorbin 1291
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 1283 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 imnan 634 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32biimpri 128 . . . 4 (¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
43adantl 266 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (𝜑 → ¬ 𝜓))
51, 4sylbi 118 . 2 ((𝜑𝜓) → (𝜑 → ¬ 𝜓))
6 pm2.53 651 . . . . 5 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 659 . . . 4 ((𝜑𝜓) → (¬ 𝜓𝜑))
87adantr 265 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (¬ 𝜓𝜑))
91, 8sylbi 118 . 2 ((𝜑𝜓) → (¬ 𝜓𝜑))
105, 9impbid 124 1 ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  wxo 1282
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 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-xor 1283
This theorem is referenced by:  xornbi  1293  zeo4  10181  odd2np1  10184
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