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Theorem condc 823
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.) (Proof shortened by BJ, 18-Nov-2023.)

Assertion
Ref Expression
condc (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))

Proof of Theorem condc
StepHypRef Expression
1 dcstab 814 . 2 (DECID 𝜑STAB 𝜑)
2 const 822 . 2 (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
31, 2syl 14 1 (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 800  DECID wdc 804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805
This theorem is referenced by:  pm2.18dc  825  con1dc  826  con4biddc  827  pm2.521gdc  838  pm2.521dcALT  840  con34bdc  841  necon4aidc  2353  necon4addc  2355  necon4bddc  2356  necon4ddc  2357  nn0n0n1ge2b  9098  gcdeq0  11592  lcmeq0  11679
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