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Theorem pm4.83dc 857
Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 761, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm4.83dc (DECID φ → (((φψ) φψ)) ↔ ψ))

Proof of Theorem pm4.83dc
StepHypRef Expression
1 df-dc 742 . . 3 (DECID φ ↔ (φ ¬ φ))
2 pm3.44 634 . . . 4 (((φψ) φψ)) → ((φ ¬ φ) → ψ))
32com12 27 . . 3 ((φ ¬ φ) → (((φψ) φψ)) → ψ))
41, 3sylbi 114 . 2 (DECID φ → (((φψ) φψ)) → ψ))
5 ax-1 5 . . 3 (ψ → (φψ))
6 ax-1 5 . . 3 (ψ → (¬ φψ))
75, 6jca 290 . 2 (ψ → ((φψ) φψ)))
84, 7impbid1 130 1 (DECID φ → (((φψ) φψ)) ↔ 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-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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