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Theorem condc 788
 Description: Contraposition of a decidable proposition. This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (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 782 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 5 . . . 4 (𝜑 → (𝜓𝜑))
32a1d 22 . . 3 (𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
4 pm2.27 40 . . . 4 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓))
5 ax-in2 581 . . . 4 𝜓 → (𝜓𝜑))
64, 5syl6 33 . . 3 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
73, 6jaoi 672 . 2 ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
81, 7sylbi 120 1 (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 665  DECID wdc 781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 581  ax-io 666 This theorem depends on definitions:  df-bi 116  df-dc 782 This theorem is referenced by:  pm2.18dc  789  con1dc  792  con4biddc  793  pm2.521dc  803  con34bdc  804  necon4aidc  2324  necon4addc  2326  necon4bddc  2327  necon4ddc  2328  nn0n0n1ge2b  8880  gcdeq0  11300  lcmeq0  11385
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