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Theorem pm5.15dc 1277
Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
pm5.15dc (DECID φ → (DECID ψ → ((φψ) (φ ↔ ¬ ψ))))

Proof of Theorem pm5.15dc
StepHypRef Expression
1 xor3dc 1275 . . . . 5 (DECID φ → (DECID ψ → (¬ (φψ) ↔ (φ ↔ ¬ ψ))))
21imp 115 . . . 4 ((DECID φ DECID ψ) → (¬ (φψ) ↔ (φ ↔ ¬ ψ)))
32biimpd 132 . . 3 ((DECID φ DECID ψ) → (¬ (φψ) → (φ ↔ ¬ ψ)))
4 dcbi 843 . . . . 5 (DECID φ → (DECID ψDECID (φψ)))
54imp 115 . . . 4 ((DECID φ DECID ψ) → DECID (φψ))
6 dfordc 790 . . . 4 (DECID (φψ) → (((φψ) (φ ↔ ¬ ψ)) ↔ (¬ (φψ) → (φ ↔ ¬ ψ))))
75, 6syl 14 . . 3 ((DECID φ DECID ψ) → (((φψ) (φ ↔ ¬ ψ)) ↔ (¬ (φψ) → (φ ↔ ¬ ψ))))
83, 7mpbird 156 . 2 ((DECID φ DECID ψ) → ((φψ) (φ ↔ ¬ ψ)))
98ex 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   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: (None)
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