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Mirrors > Home > ILE Home > Th. List > condandc | Unicode version |
Description: Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume , derive a contradiction, and therefore conclude . By contrast, assuming , deriving a contradiction, and therefore concluding , as in pm2.65 585, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
Ref | Expression |
---|---|
condandc.1 | |
condandc.2 |
Ref | Expression |
---|---|
condandc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | condandc.1 | . . 3 | |
2 | condandc.2 | . . 3 | |
3 | 1, 2 | pm2.65da 587 | . 2 |
4 | notnotrdc 751 | . 2 DECID | |
5 | 3, 4 | syl5 28 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: (None) |
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