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Theorem xornbidc 1265
Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
Assertion
Ref Expression
xornbidc (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φψ))))

Proof of Theorem xornbidc
StepHypRef Expression
1 xor2dc 1264 . . . 4 (DECID φ → (DECID ψ → (¬ (φψ) ↔ ((φ ψ) ¬ (φ ψ)))))
21imp 115 . . 3 ((DECID φ DECID ψ) → (¬ (φψ) ↔ ((φ ψ) ¬ (φ ψ))))
3 df-xor 1252 . . 3 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
42, 3syl6rbbr 188 . 2 ((DECID φ DECID ψ) → ((φψ) ↔ ¬ (φψ)))
54ex 108 1 (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616  DECID wdc 733  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-dc 734  df-xor 1252
This theorem is referenced by:  xordc  1266  xordidc  1273
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