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Theorem condc 760
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 754 . 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 555 . . . 4 𝜓 → (𝜓𝜑))
64, 5syl6 33 . . 3 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
73, 6jaoi 646 . 2 ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
81, 7sylbi 118 1 (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm2.18dc  761  con1dc  764  con4biddc  765  pm2.521dc  775  con34bdc  776  necon4aidc  2288  necon4addc  2290  necon4bddc  2291  necon4ddc  2292  nn0n0n1ge2b  8378
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