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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 845 | . 2 ⊢ (DECID 𝜑 → STAB 𝜑) | |
2 | const 853 | . 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 831 DECID wdc 835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 |
This theorem is referenced by: pm2.18dc 856 con1dc 857 con4biddc 858 pm2.521gdc 869 pm2.521dcALT 871 con34bdc 872 necon4aidc 2425 necon4addc 2427 necon4bddc 2428 necon4ddc 2429 nn0n0n1ge2b 9346 gcdeq0 11992 lcmeq0 12085 pcdvdsb 12333 pc2dvds 12343 pcfac 12362 infpnlem1 12371 m1lgs 14748 |
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