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Theorem xor3dc 1275
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
xor3dc (DECID φ → (DECID ψ → (¬ (φψ) ↔ (φ ↔ ¬ ψ))))

Proof of Theorem xor3dc
StepHypRef Expression
1 dcn 745 . . . . . 6 (DECID ψDECID ¬ ψ)
2 dcbi 843 . . . . . 6 (DECID φ → (DECID ¬ ψDECID (φ ↔ ¬ ψ)))
31, 2syl5 28 . . . . 5 (DECID φ → (DECID ψDECID (φ ↔ ¬ ψ)))
43imp 115 . . . 4 ((DECID φ DECID ψ) → DECID (φ ↔ ¬ ψ))
5 pm5.18dc 776 . . . . . . 7 (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
65imp 115 . . . . . 6 ((DECID φ DECID ψ) → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
76a1d 22 . . . . 5 ((DECID φ DECID ψ) → (DECID (φ ↔ ¬ ψ) → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
87con2biddc 773 . . . 4 ((DECID φ DECID ψ) → (DECID (φ ↔ ¬ ψ) → ((φ ↔ ¬ ψ) ↔ ¬ (φψ))))
94, 8mpd 13 . . 3 ((DECID φ DECID ψ) → ((φ ↔ ¬ ψ) ↔ ¬ (φψ)))
109bicomd 129 . 2 ((DECID φ DECID ψ) → (¬ (φψ) ↔ (φ ↔ ¬ ψ)))
1110ex 108 1 (DECID φ → (DECID ψ → (¬ (φψ) ↔ (φ ↔ ¬ ψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  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:  pm5.15dc  1277  xor2dc  1278  nbbndc  1282
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