Theorem condc 782

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.)

Assertion

Ref	Expression
condc	⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Proof of Theorem condc

Step	Hyp	Ref	Expression
1		df-dc 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 5	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜑))
3	2	a1d 22	. . 3 ⊢ (𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
4		pm2.27 39	. . . 4 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓))
5		ax-in2 577	. . . 4 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
6	4, 5	syl6 33	. . 3 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
7	3, 6	jaoi 668	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
8	1, 7	sylbi 119	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm2.18dc  783  con1dc  786  con4biddc  787  pm2.521dc  797  con34bdc  798  necon4aidc  2313  necon4addc  2315  necon4bddc  2316  necon4ddc  2317  nn0n0n1ge2b  8427  gcdeq0  10368  lcmeq0  10453