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Mirrors > Home > ILE Home > Th. List > dcn | GIF version |
Description: A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
Ref | Expression |
---|---|
dcn | ⊢ (DECID φ → DECID ¬ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnot1 559 | . . . 4 ⊢ (φ → ¬ ¬ φ) | |
2 | 1 | orim2i 677 | . . 3 ⊢ ((¬ φ ∨ φ) → (¬ φ ∨ ¬ ¬ φ)) |
3 | 2 | orcoms 648 | . 2 ⊢ ((φ ∨ ¬ φ) → (¬ φ ∨ ¬ ¬ φ)) |
4 | df-dc 742 | . 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ)) | |
5 | df-dc 742 | . 2 ⊢ (DECID ¬ φ ↔ (¬ φ ∨ ¬ ¬ φ)) | |
6 | 3, 4, 5 | 3imtr4i 190 | 1 ⊢ (DECID φ → DECID ¬ φ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 628 DECID wdc 741 |
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 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: pm5.18dc 776 pm4.67dc 780 pm2.54dc 789 imordc 795 pm4.54dc 804 stabtestimpdc 823 annimdc 844 pm4.55dc 845 pm3.12dc 864 pm3.13dc 865 dn1dc 866 xor3dc 1275 dfbi3dc 1285 dcned 2209 |
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