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Theorem condc 720
Description: Contraposition of a decidable proposition. (Contributed by Jim Kingdon, 13-Mar-2018.)
Assertion
Ref Expression
condc (DECID φ → ((¬ φ → ¬ ψ) → (ψφ)))

Proof of Theorem condc
StepHypRef Expression
1 df-dc 715 . 2 (DECID φ ↔ (φ ¬ φ))
2 ax-1 5 . . . 4 (φ → (ψφ))
32a1d 20 . . 3 (φ → ((¬ φ → ¬ ψ) → (ψφ)))
4 pm2.27 33 . . . 4 φ → ((¬ φ → ¬ ψ) → ¬ ψ))
5 ax-in2 527 . . . 4 ψ → (ψφ))
64, 5syl6 27 . . 3 φ → ((¬ φ → ¬ ψ) → (ψφ)))
73, 6jaoi 612 . 2 ((φ ¬ φ) → ((¬ φ → ¬ ψ) → (ψφ)))
81, 7sylbi 112 1 (DECID φ → ((¬ φ → ¬ ψ) → (ψφ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 605  DECID wdc 714
This theorem is referenced by:  pm2.18dc  723  con1dc  728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in2 527  ax-io 606
This theorem depends on definitions:  df-bi 108  df-dc 715
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