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Mirrors > Home > ILE Home > Th. List > dfandc | Unicode version |
Description: Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 637. (Contributed by Jim Kingdon, 30-Apr-2018.) |
Ref | Expression |
---|---|
dfandc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2im 637 |
. . . 4
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2 | 1 | imp 124 |
. . 3
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3 | simplimdc 860 |
. . . . . . 7
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4 | 3 | adantr 276 |
. . . . . 6
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5 | 4 | imp 124 |
. . . . 5
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6 | simprimdc 859 |
. . . . . . 7
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7 | 6 | adantl 277 |
. . . . . 6
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8 | 7 | imp 124 |
. . . . 5
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9 | 5, 8 | jca 306 |
. . . 4
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10 | 9 | ex 115 |
. . 3
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11 | 2, 10 | impbid2 143 |
. 2
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12 | 11 | ex 115 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 |
This theorem is referenced by: pm4.63dc 886 pm4.54dc 902 |
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