Theorem condc 865

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 856	. 2 ⊢ (DECID 𝜑 → STAB 𝜑)
2		const 864	. 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 842  DECID wdc 846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is referenced by:  pm2.18dc  867  con1dc  868  con4biddc  869  pm2.521gdc  880  pm2.521dcALT  882  con34bdc  883  necon4aidc  2488  necon4addc  2490  necon4bddc  2491  necon4ddc  2492  nn0n0n1ge2b  9708  gcdeq0  12737  lcmeq0  12832  pcdvdsb  13082  pc2dvds  13092  pcfac  13112  infpnlem1  13121  m1lgs  16187  exmidcon  17019