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Mirrors > Home > ILE Home > Th. List > ax16i | Unicode version |
Description: Inference with ax-16 1787 as its conclusion, that doesn't require ax-10 1484, ax-11 1485, or ax-12 1490 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.) |
Ref | Expression |
---|---|
ax16i.1 |
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ax16i.2 |
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Ref | Expression |
---|---|
ax16i |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1507 |
. . . 4
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2 | ax-17 1507 |
. . . 4
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3 | ax-8 1483 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | cbv3h 1722 |
. . 3
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5 | ax-8 1483 |
. . . . . 6
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6 | 5 | spimv 1784 |
. . . . 5
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7 | equid 1678 |
. . . . . . . 8
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8 | ax-8 1483 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 7, 8 | mpi 15 |
. . . . . . 7
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10 | equid 1678 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
11 | ax-8 1483 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | mpi 15 |
. . . . . . . 8
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13 | ax-8 1483 |
. . . . . . . 8
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14 | 12, 13 | syl 14 |
. . . . . . 7
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15 | 9, 14 | syl5com 29 |
. . . . . 6
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16 | 1, 15 | alimdh 1444 |
. . . . 5
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17 | 6, 16 | mpcom 36 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | ax-8 1483 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 10, 18 | mpi 15 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | alimi 1432 |
. . . 4
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21 | 17, 20 | syl 14 |
. . 3
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22 | 4, 21 | syl 14 |
. 2
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23 | ax-17 1507 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | ax16i.1 |
. . . . 5
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25 | 24 | biimpcd 158 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 23, 25 | alimdh 1444 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | ax16i.2 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 24 | biimprd 157 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 19, 28 | syl 14 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 27, 23, 29 | cbv3h 1722 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 26, 30 | syl6com 35 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 22, 31 | syl 14 |
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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: ax16ALT 1832 |
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