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Theorem xorbin 1275
 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 1267 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 imnan 624 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32biimpri 124 . . . 4 (¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
43adantl 262 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (𝜑 → ¬ 𝜓))
51, 4sylbi 114 . 2 ((𝜑𝜓) → (𝜑 → ¬ 𝜓))
6 pm2.53 641 . . . . 5 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 649 . . . 4 ((𝜑𝜓) → (¬ 𝜓𝜑))
87adantr 261 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (¬ 𝜓𝜑))
91, 8sylbi 114 . 2 ((𝜑𝜓) → (¬ 𝜓𝜑))
105, 9impbid 120 1 ((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ⊻ wxo 1266 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-xor 1267 This theorem is referenced by:  xornbi  1277
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