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Mirrors > Home > ILE Home > Th. List > preq12bg | Unicode version |
Description: Closed form of preq12b 3562. (Contributed by Scott Fenton, 28-Mar-2014.) |
Ref | Expression |
---|---|
preq12bg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3469 | . . . . . . 7 | |
2 | 1 | eqeq1d 2089 | . . . . . 6 |
3 | eqeq1 2087 | . . . . . . . 8 | |
4 | 3 | anbi1d 452 | . . . . . . 7 |
5 | eqeq1 2087 | . . . . . . . 8 | |
6 | 5 | anbi1d 452 | . . . . . . 7 |
7 | 4, 6 | orbi12d 739 | . . . . . 6 |
8 | 2, 7 | bibi12d 233 | . . . . 5 |
9 | 8 | imbi2d 228 | . . . 4 |
10 | preq2 3470 | . . . . . . 7 | |
11 | 10 | eqeq1d 2089 | . . . . . 6 |
12 | eqeq1 2087 | . . . . . . . 8 | |
13 | 12 | anbi2d 451 | . . . . . . 7 |
14 | eqeq1 2087 | . . . . . . . 8 | |
15 | 14 | anbi2d 451 | . . . . . . 7 |
16 | 13, 15 | orbi12d 739 | . . . . . 6 |
17 | 11, 16 | bibi12d 233 | . . . . 5 |
18 | 17 | imbi2d 228 | . . . 4 |
19 | preq1 3469 | . . . . . . 7 | |
20 | 19 | eqeq2d 2092 | . . . . . 6 |
21 | eqeq2 2090 | . . . . . . . 8 | |
22 | 21 | anbi1d 452 | . . . . . . 7 |
23 | eqeq2 2090 | . . . . . . . 8 | |
24 | 23 | anbi2d 451 | . . . . . . 7 |
25 | 22, 24 | orbi12d 739 | . . . . . 6 |
26 | 20, 25 | bibi12d 233 | . . . . 5 |
27 | 26 | imbi2d 228 | . . . 4 |
28 | preq2 3470 | . . . . . . 7 | |
29 | 28 | eqeq2d 2092 | . . . . . 6 |
30 | eqeq2 2090 | . . . . . . . 8 | |
31 | 30 | anbi2d 451 | . . . . . . 7 |
32 | eqeq2 2090 | . . . . . . . 8 | |
33 | 32 | anbi1d 452 | . . . . . . 7 |
34 | 31, 33 | orbi12d 739 | . . . . . 6 |
35 | vex 2604 | . . . . . . 7 | |
36 | vex 2604 | . . . . . . 7 | |
37 | vex 2604 | . . . . . . 7 | |
38 | vex 2604 | . . . . . . 7 | |
39 | 35, 36, 37, 38 | preq12b 3562 | . . . . . 6 |
40 | 29, 34, 39 | vtoclbg 2659 | . . . . 5 |
41 | 40 | a1i 9 | . . . 4 |
42 | 9, 18, 27, 41 | vtocl3ga 2668 | . . 3 |
43 | 42 | 3expa 1138 | . 2 |
44 | 43 | impr 371 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: prneimg 3566 |
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