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Theorem xorbin 1258
 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 1252 . . 3 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
2 imnan 611 . . . . 5 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
32biimpri 124 . . . 4 (¬ (φ ψ) → (φ → ¬ ψ))
43adantl 262 . . 3 (((φ ψ) ¬ (φ ψ)) → (φ → ¬ ψ))
51, 4sylbi 114 . 2 ((φψ) → (φ → ¬ ψ))
6 pm2.53 628 . . . . 5 ((ψ φ) → (¬ ψφ))
76orcoms 636 . . . 4 ((φ ψ) → (¬ ψφ))
87adantr 261 . . 3 (((φ ψ) ¬ (φ ψ)) → (¬ 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 616   ⊻ wxo 1251 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 532  ax-in2 533  ax-io 617 This theorem depends on definitions:  df-bi 110  df-xor 1252 This theorem is referenced by:  xornbi  1260
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