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Mirrors > Home > ILE Home > Th. List > annimdc | Unicode version |
Description: Express conjunction in terms of implication. The forward direction, annimim 676, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.) |
Ref | Expression |
---|---|
annimdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imandc 875 |
. . . 4
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2 | 1 | adantl 275 |
. . 3
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3 | dcim 827 |
. . . . 5
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4 | 3 | imp 123 |
. . . 4
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5 | dcn 828 |
. . . . . 6
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6 | dcan 919 |
. . . . . 6
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7 | 5, 6 | syl5 32 |
. . . . 5
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8 | 7 | imp 123 |
. . . 4
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9 | con2bidc 861 |
. . . 4
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10 | 4, 8, 9 | sylc 62 |
. . 3
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11 | 2, 10 | mpbid 146 |
. 2
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12 | 11 | ex 114 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
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 817 df-dc 821 |
This theorem is referenced by: xordidc 1378 |
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