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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elim2ifim | Structured version Visualization version Unicode version |
Description: Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
elim2if.1 |
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elim2if.2 |
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elim2if.3 |
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elim2ifim.1 |
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elim2ifim.2 |
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elim2ifim.3 |
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Ref | Expression |
---|---|
elim2ifim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid 421 |
. . 3
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2 | elim2ifim.1 |
. . . . 5
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3 | 2 | ancli 558 |
. . . 4
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4 | pm4.42 977 |
. . . . . 6
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5 | elim2ifim.2 |
. . . . . . . . . 10
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6 | 5 | ex 440 |
. . . . . . . . 9
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7 | 6 | ancld 560 |
. . . . . . . 8
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8 | 7 | imp 435 |
. . . . . . 7
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9 | elim2ifim.3 |
. . . . . . . . . 10
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10 | 9 | ex 440 |
. . . . . . . . 9
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11 | 10 | ancld 560 |
. . . . . . . 8
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12 | 11 | imp 435 |
. . . . . . 7
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13 | 8, 12 | orim12i 523 |
. . . . . 6
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14 | 4, 13 | sylbi 200 |
. . . . 5
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15 | 14 | ancli 558 |
. . . 4
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16 | 3, 15 | orim12i 523 |
. . 3
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17 | 1, 16 | ax-mp 5 |
. 2
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18 | elim2if.1 |
. . 3
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19 | elim2if.2 |
. . 3
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20 | elim2if.3 |
. . 3
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21 | 18, 19, 20 | elim2if 28209 |
. 2
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22 | 17, 21 | mpbir 214 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-if 3893 |
This theorem is referenced by: (None) |
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