Theorem condc 860

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 851	. 2 ⊢ (DECID 𝜑 → STAB 𝜑)
2		const 859	. 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 837  DECID wdc 841

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 619  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842

This theorem is referenced by:  pm2.18dc  862  con1dc  863  con4biddc  864  pm2.521gdc  875  pm2.521dcALT  877  con34bdc  878  necon4aidc  2470  necon4addc  2472  necon4bddc  2473  necon4ddc  2474  nn0n0n1ge2b  9559  gcdeq0  12550  lcmeq0  12645  pcdvdsb  12895  pc2dvds  12905  pcfac  12925  infpnlem1  12934  m1lgs  15817