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Mirrors > Home > ILE Home > Th. List > preq12b | Unicode version |
Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.) |
Ref | Expression |
---|---|
preq12b.1 |
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preq12b.2 |
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preq12b.3 |
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preq12b.4 |
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Ref | Expression |
---|---|
preq12b |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 |
. . . . . 6
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2 | 1 | prid1 3568 |
. . . . 5
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3 | eleq2 2158 |
. . . . 5
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4 | 2, 3 | mpbii 147 |
. . . 4
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5 | 1 | elpr 3487 |
. . . 4
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6 | 4, 5 | sylib 121 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | preq1 3539 |
. . . . . . . 8
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8 | 7 | eqeq1d 2103 |
. . . . . . 7
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9 | preq12b.2 |
. . . . . . . 8
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10 | preq12b.4 |
. . . . . . . 8
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11 | 9, 10 | preqr2 3635 |
. . . . . . 7
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12 | 8, 11 | syl6bi 162 |
. . . . . 6
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13 | 12 | com12 30 |
. . . . 5
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14 | 13 | ancld 319 |
. . . 4
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15 | prcom 3538 |
. . . . . . 7
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16 | 15 | eqeq2i 2105 |
. . . . . 6
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17 | preq1 3539 |
. . . . . . . . 9
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18 | 17 | eqeq1d 2103 |
. . . . . . . 8
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19 | preq12b.3 |
. . . . . . . . 9
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20 | 9, 19 | preqr2 3635 |
. . . . . . . 8
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21 | 18, 20 | syl6bi 162 |
. . . . . . 7
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22 | 21 | com12 30 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 16, 22 | sylbi 120 |
. . . . 5
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24 | 23 | ancld 319 |
. . . 4
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25 | 14, 24 | orim12d 738 |
. . 3
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26 | 6, 25 | mpd 13 |
. 2
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27 | preq12 3541 |
. . 3
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28 | prcom 3538 |
. . . . 5
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29 | 17, 28 | syl6eq 2143 |
. . . 4
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30 | preq1 3539 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | sylan9eq 2147 |
. . 3
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32 | 27, 31 | jaoi 674 |
. 2
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33 | 26, 32 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 |
This theorem is referenced by: prel12 3637 opthpr 3638 preq12bg 3639 preqsn 3641 opeqpr 4104 preleq 4399 |
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