Theorem condc 839

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) → (𝜓 → 𝜑)))

Proof of Theorem condc

Step	Hyp	Ref	Expression
1		dcstab 830	. 2 ⊢ (DECID 𝜑 → STAB 𝜑)
2		const 838	. 2 ⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
3	1, 2	syl 14	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 816  DECID wdc 820

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821

This theorem is referenced by:  pm2.18dc  841  con1dc  842  con4biddc  843  pm2.521gdc  854  pm2.521dcALT  856  con34bdc  857  necon4aidc  2377  necon4addc  2379  necon4bddc  2380  necon4ddc  2381  nn0n0n1ge2b  9154  gcdeq0  11701  lcmeq0  11788