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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.) (Proof shortened by BJ, 18-Nov-2023.) |
Ref | Expression |
---|---|
condc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcstab 814 | . 2 ⊢ (DECID 𝜑 → STAB 𝜑) | |
2 | const 822 | . 2 ⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) | |
3 | 1, 2 | syl 14 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 STAB wstab 800 DECID wdc 804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 |
This theorem is referenced by: pm2.18dc 825 con1dc 826 con4biddc 827 pm2.521gdc 838 pm2.521dcALT 840 con34bdc 841 necon4aidc 2353 necon4addc 2355 necon4bddc 2356 necon4ddc 2357 nn0n0n1ge2b 9098 gcdeq0 11592 lcmeq0 11679 |
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