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Mirrors > Home > ILE Home > Th. List > preq12bg | Unicode version |
Description: Closed form of preq12b 3697. (Contributed by Scott Fenton, 28-Mar-2014.) |
Ref | Expression |
---|---|
preq12bg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3600 | . . . . . . 7 | |
2 | 1 | eqeq1d 2148 | . . . . . 6 |
3 | eqeq1 2146 | . . . . . . . 8 | |
4 | 3 | anbi1d 460 | . . . . . . 7 |
5 | eqeq1 2146 | . . . . . . . 8 | |
6 | 5 | anbi1d 460 | . . . . . . 7 |
7 | 4, 6 | orbi12d 782 | . . . . . 6 |
8 | 2, 7 | bibi12d 234 | . . . . 5 |
9 | 8 | imbi2d 229 | . . . 4 |
10 | preq2 3601 | . . . . . . 7 | |
11 | 10 | eqeq1d 2148 | . . . . . 6 |
12 | eqeq1 2146 | . . . . . . . 8 | |
13 | 12 | anbi2d 459 | . . . . . . 7 |
14 | eqeq1 2146 | . . . . . . . 8 | |
15 | 14 | anbi2d 459 | . . . . . . 7 |
16 | 13, 15 | orbi12d 782 | . . . . . 6 |
17 | 11, 16 | bibi12d 234 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | preq1 3600 | . . . . . . 7 | |
20 | 19 | eqeq2d 2151 | . . . . . 6 |
21 | eqeq2 2149 | . . . . . . . 8 | |
22 | 21 | anbi1d 460 | . . . . . . 7 |
23 | eqeq2 2149 | . . . . . . . 8 | |
24 | 23 | anbi2d 459 | . . . . . . 7 |
25 | 22, 24 | orbi12d 782 | . . . . . 6 |
26 | 20, 25 | bibi12d 234 | . . . . 5 |
27 | 26 | imbi2d 229 | . . . 4 |
28 | preq2 3601 | . . . . . . 7 | |
29 | 28 | eqeq2d 2151 | . . . . . 6 |
30 | eqeq2 2149 | . . . . . . . 8 | |
31 | 30 | anbi2d 459 | . . . . . . 7 |
32 | eqeq2 2149 | . . . . . . . 8 | |
33 | 32 | anbi1d 460 | . . . . . . 7 |
34 | 31, 33 | orbi12d 782 | . . . . . 6 |
35 | vex 2689 | . . . . . . 7 | |
36 | vex 2689 | . . . . . . 7 | |
37 | vex 2689 | . . . . . . 7 | |
38 | vex 2689 | . . . . . . 7 | |
39 | 35, 36, 37, 38 | preq12b 3697 | . . . . . 6 |
40 | 29, 34, 39 | vtoclbg 2747 | . . . . 5 |
41 | 40 | a1i 9 | . . . 4 |
42 | 9, 18, 27, 41 | vtocl3ga 2756 | . . 3 |
43 | 42 | 3expa 1181 | . 2 |
44 | 43 | impr 376 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 cpr 3528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 |
This theorem is referenced by: prneimg 3701 |
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