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Mirrors > Home > ILE Home > Th. List > preq12bg | Unicode version |
Description: Closed form of preq12b 3541. (Contributed by Scott Fenton, 28-Mar-2014.) |
Ref | Expression |
---|---|
preq12bg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3447 | . . . . . . 7 | |
2 | 1 | eqeq1d 2048 | . . . . . 6 |
3 | eqeq1 2046 | . . . . . . . 8 | |
4 | 3 | anbi1d 438 | . . . . . . 7 |
5 | eqeq1 2046 | . . . . . . . 8 | |
6 | 5 | anbi1d 438 | . . . . . . 7 |
7 | 4, 6 | orbi12d 707 | . . . . . 6 |
8 | 2, 7 | bibi12d 224 | . . . . 5 |
9 | 8 | imbi2d 219 | . . . 4 |
10 | preq2 3448 | . . . . . . 7 | |
11 | 10 | eqeq1d 2048 | . . . . . 6 |
12 | eqeq1 2046 | . . . . . . . 8 | |
13 | 12 | anbi2d 437 | . . . . . . 7 |
14 | eqeq1 2046 | . . . . . . . 8 | |
15 | 14 | anbi2d 437 | . . . . . . 7 |
16 | 13, 15 | orbi12d 707 | . . . . . 6 |
17 | 11, 16 | bibi12d 224 | . . . . 5 |
18 | 17 | imbi2d 219 | . . . 4 |
19 | preq1 3447 | . . . . . . 7 | |
20 | 19 | eqeq2d 2051 | . . . . . 6 |
21 | eqeq2 2049 | . . . . . . . 8 | |
22 | 21 | anbi1d 438 | . . . . . . 7 |
23 | eqeq2 2049 | . . . . . . . 8 | |
24 | 23 | anbi2d 437 | . . . . . . 7 |
25 | 22, 24 | orbi12d 707 | . . . . . 6 |
26 | 20, 25 | bibi12d 224 | . . . . 5 |
27 | 26 | imbi2d 219 | . . . 4 |
28 | preq2 3448 | . . . . . . 7 | |
29 | 28 | eqeq2d 2051 | . . . . . 6 |
30 | eqeq2 2049 | . . . . . . . 8 | |
31 | 30 | anbi2d 437 | . . . . . . 7 |
32 | eqeq2 2049 | . . . . . . . 8 | |
33 | 32 | anbi1d 438 | . . . . . . 7 |
34 | 31, 33 | orbi12d 707 | . . . . . 6 |
35 | vex 2560 | . . . . . . 7 | |
36 | vex 2560 | . . . . . . 7 | |
37 | vex 2560 | . . . . . . 7 | |
38 | vex 2560 | . . . . . . 7 | |
39 | 35, 36, 37, 38 | preq12b 3541 | . . . . . 6 |
40 | 29, 34, 39 | vtoclbg 2614 | . . . . 5 |
41 | 40 | a1i 9 | . . . 4 |
42 | 9, 18, 27, 41 | vtocl3ga 2623 | . . 3 |
43 | 42 | 3expa 1104 | . 2 |
44 | 43 | impr 361 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wcel 1393 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-3an 887 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: prneimg 3545 |
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