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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 754 | . 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 555 | . . . 4 ⊢ (¬ 𝜓 → (𝜓 → 𝜑)) | |
6 | 4, 5 | syl6 33 | . . 3 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
7 | 3, 6 | jaoi 646 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
8 | 1, 7 | sylbi 118 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 639 DECID wdc 753 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 103 ax-ia2 104 ax-ia3 105 ax-in2 555 ax-io 640 |
This theorem depends on definitions: df-bi 114 df-dc 754 |
This theorem is referenced by: pm2.18dc 761 con1dc 764 con4biddc 765 pm2.521dc 775 con34bdc 776 necon4aidc 2288 necon4addc 2290 necon4bddc 2291 necon4ddc 2292 nn0n0n1ge2b 8378 |
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