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Mirrors > Home > ILE Home > Th. List > dcan | Unicode version |
Description: A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) |
Ref | Expression |
---|---|
dcan |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 |
. . . . . 6
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2 | 1 | intnanrd 875 |
. . . . 5
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3 | 2 | orim2i 711 |
. . . 4
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4 | simpr 108 |
. . . . . 6
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5 | 4 | intnand 874 |
. . . . 5
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6 | 5 | olcd 686 |
. . . 4
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7 | 3, 6 | jaoi 669 |
. . 3
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8 | df-dc 777 |
. . . . 5
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9 | df-dc 777 |
. . . . 5
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10 | 8, 9 | anbi12i 448 |
. . . 4
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11 | andi 765 |
. . . 4
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12 | andir 766 |
. . . . 5
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13 | 12 | orbi1i 713 |
. . . 4
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14 | 10, 11, 13 | 3bitri 204 |
. . 3
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15 | df-dc 777 |
. . 3
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16 | 7, 14, 15 | 3imtr4i 199 |
. 2
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17 | 16 | ex 113 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 |
This theorem depends on definitions: df-bi 115 df-dc 777 |
This theorem is referenced by: dcbi 878 annimdc 879 pm4.55dc 880 orandc 881 anordc 898 xordidc 1331 nn0n0n1ge2b 8497 gcdmndc 10473 gcdsupex 10482 gcdsupcl 10483 gcdaddm 10508 lcmval 10578 lcmcllem 10582 lcmledvds 10585 |
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