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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 782 | . 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 40 | . . . 4 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓)) | |
5 | ax-in2 581 | . . . 4 ⊢ (¬ 𝜓 → (𝜓 → 𝜑)) | |
6 | 4, 5 | syl6 33 | . . 3 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
7 | 3, 6 | jaoi 672 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
8 | 1, 7 | sylbi 120 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 665 DECID wdc 781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 581 ax-io 666 |
This theorem depends on definitions: df-bi 116 df-dc 782 |
This theorem is referenced by: pm2.18dc 789 con1dc 792 con4biddc 793 pm2.521dc 803 con34bdc 804 necon4aidc 2324 necon4addc 2326 necon4bddc 2327 necon4ddc 2328 nn0n0n1ge2b 8880 gcdeq0 11300 lcmeq0 11385 |
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