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Mirrors > Home > ILE Home > Th. List > condc | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcstab 834 | . 2 DECID STAB | |
2 | const 842 | . 2 STAB | |
3 | 1, 2 | syl 14 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 STAB wstab 820 DECID wdc 824 |
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 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 |
This theorem is referenced by: pm2.18dc 845 con1dc 846 con4biddc 847 pm2.521gdc 858 pm2.521dcALT 860 con34bdc 861 necon4aidc 2402 necon4addc 2404 necon4bddc 2405 necon4ddc 2406 nn0n0n1ge2b 9262 gcdeq0 11899 lcmeq0 11992 pcdvdsb 12240 pc2dvds 12250 pcfac 12269 infpnlem1 12278 |
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