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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 | |
preq12b.2 | |
preq12b.3 | |
preq12b.4 |
Ref | Expression |
---|---|
preq12b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 | . . . . . 6 | |
2 | 1 | prid1 3476 | . . . . 5 |
3 | eleq2 2101 | . . . . 5 | |
4 | 2, 3 | mpbii 136 | . . . 4 |
5 | 1 | elpr 3396 | . . . 4 |
6 | 4, 5 | sylib 127 | . . 3 |
7 | preq1 3447 | . . . . . . . 8 | |
8 | 7 | eqeq1d 2048 | . . . . . . 7 |
9 | preq12b.2 | . . . . . . . 8 | |
10 | preq12b.4 | . . . . . . . 8 | |
11 | 9, 10 | preqr2 3540 | . . . . . . 7 |
12 | 8, 11 | syl6bi 152 | . . . . . 6 |
13 | 12 | com12 27 | . . . . 5 |
14 | 13 | ancld 308 | . . . 4 |
15 | prcom 3446 | . . . . . . 7 | |
16 | 15 | eqeq2i 2050 | . . . . . 6 |
17 | preq1 3447 | . . . . . . . . 9 | |
18 | 17 | eqeq1d 2048 | . . . . . . . 8 |
19 | preq12b.3 | . . . . . . . . 9 | |
20 | 9, 19 | preqr2 3540 | . . . . . . . 8 |
21 | 18, 20 | syl6bi 152 | . . . . . . 7 |
22 | 21 | com12 27 | . . . . . 6 |
23 | 16, 22 | sylbi 114 | . . . . 5 |
24 | 23 | ancld 308 | . . . 4 |
25 | 14, 24 | orim12d 700 | . . 3 |
26 | 6, 25 | mpd 13 | . 2 |
27 | preq12 3449 | . . 3 | |
28 | prcom 3446 | . . . . 5 | |
29 | 17, 28 | syl6eq 2088 | . . . 4 |
30 | preq1 3447 | . . . 4 | |
31 | 29, 30 | sylan9eq 2092 | . . 3 |
32 | 27, 31 | jaoi 636 | . 2 |
33 | 26, 32 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 cvv 2557 cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: prel12 3542 opthpr 3543 preq12bg 3544 preqsn 3546 opeqpr 3990 preleq 4279 |
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