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Mirrors > Home > ILE Home > Th. List > condc | GIF version |
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.) |
Ref | Expression |
---|---|
condc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 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 35 | . . . 4 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓)) | |
5 | ax-in2 545 | . . . 4 ⊢ (¬ 𝜓 → (𝜓 → 𝜑)) | |
6 | 4, 5 | syl6 29 | . . 3 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
7 | 3, 6 | jaoi 636 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
8 | 1, 7 | sylbi 114 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: pm2.18dc 750 con1dc 753 con4biddc 754 pm2.521dc 764 con34bdc 765 necon4aidc 2273 necon4addc 2275 necon4bddc 2276 necon4ddc 2277 nn0n0n1ge2b 8320 |
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