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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 3713 |
. . . . 5
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3 | eleq2 2253 |
. . . . 5
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4 | 2, 3 | mpbii 148 |
. . . 4
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5 | 1 | elpr 3628 |
. . . 4
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6 | 4, 5 | sylib 122 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | preq1 3684 |
. . . . . . . 8
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8 | 7 | eqeq1d 2198 |
. . . . . . 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 3784 |
. . . . . . 7
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12 | 8, 11 | biimtrdi 163 |
. . . . . 6
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13 | 12 | com12 30 |
. . . . 5
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14 | 13 | ancld 325 |
. . . 4
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15 | prcom 3683 |
. . . . . . 7
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16 | 15 | eqeq2i 2200 |
. . . . . 6
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17 | preq1 3684 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | eqeq1d 2198 |
. . . . . . . 8
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19 | preq12b.3 |
. . . . . . . . 9
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20 | 9, 19 | preqr2 3784 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 18, 20 | biimtrdi 163 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | com12 30 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 16, 22 | sylbi 121 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | ancld 325 |
. . . 4
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25 | 14, 24 | orim12d 787 |
. . 3
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26 | 6, 25 | mpd 13 |
. 2
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27 | preq12 3686 |
. . 3
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28 | prcom 3683 |
. . . . 5
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29 | 17, 28 | eqtrdi 2238 |
. . . 4
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30 | preq1 3684 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | sylan9eq 2242 |
. . 3
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32 | 27, 31 | jaoi 717 |
. 2
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33 | 26, 32 | impbii 126 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 |
This theorem is referenced by: prel12 3786 opthpr 3787 preq12bg 3788 preqsn 3790 opeqpr 4271 preleq 4572 |
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