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Theorem xor2dc 1278
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
xor2dc (DECID φ → (DECID ψ → (¬ (φψ) ↔ ((φ ψ) ¬ (φ ψ)))))

Proof of Theorem xor2dc
StepHypRef Expression
1 xor3dc 1275 . . . 4 (DECID φ → (DECID ψ → (¬ (φψ) ↔ (φ ↔ ¬ ψ))))
21imp 115 . . 3 ((DECID φ DECID ψ) → (¬ (φψ) ↔ (φ ↔ ¬ ψ)))
3 pm5.17dc 809 . . . 4 (DECID ψ → (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ)))
43adantl 262 . . 3 ((DECID φ DECID ψ) → (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ)))
52, 4bitr4d 180 . 2 ((DECID φ DECID ψ) → (¬ (φψ) ↔ ((φ ψ) ¬ (φ ψ))))
65ex 108 1 (DECID φ → (DECID ψ → (¬ (φψ) ↔ ((φ ψ) ¬ (φ ψ)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  xornbidc  1279
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