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Theorem condc 783
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 777 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ax-1 5 . . . 4 (𝜑 → (𝜓𝜑))
32a1d 22 . . 3 (𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
4 pm2.27 39 . . . 4 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓))
5 ax-in2 578 . . . 4 𝜓 → (𝜓𝜑))
64, 5syl6 33 . . 3 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
73, 6jaoi 669 . 2 ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
81, 7sylbi 119 1 (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  pm2.18dc  784  con1dc  787  con4biddc  788  pm2.521dc  798  con34bdc  799  necon4aidc  2317  necon4addc  2319  necon4bddc  2320  necon4ddc  2321  nn0n0n1ge2b  8722  gcdeq0  10748  lcmeq0  10833
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