Theorem condc 843

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 834	. 2 ⊢ (DECID 𝜑 → STAB 𝜑)
2		const 842	. 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 820  DECID wdc 824

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 821  df-dc 825

This theorem is referenced by:  pm2.18dc  845  con1dc  846  con4biddc  847  pm2.521gdc  858  pm2.521dcALT  860  con34bdc  861  necon4aidc  2403  necon4addc  2405  necon4bddc  2406  necon4ddc  2407  nn0n0n1ge2b  9266  gcdeq0  11906  lcmeq0  11999  pcdvdsb  12247  pc2dvds  12257  pcfac  12276  infpnlem1  12285