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Mirrors > Home > ILE Home > Th. List > dfoprab4f | Unicode version |
Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfoprab4f.x | |
dfoprab4f.y | |
dfoprab4f.1 |
Ref | Expression |
---|---|
dfoprab4f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . . . . 5 | |
2 | dfoprab4f.x | . . . . . 6 | |
3 | nfs1v 1815 | . . . . . 6 | |
4 | 2, 3 | nfbi 1481 | . . . . 5 |
5 | 1, 4 | nfim 1464 | . . . 4 |
6 | opeq1 3549 | . . . . . 6 | |
7 | 6 | eqeq2d 2051 | . . . . 5 |
8 | sbequ12 1654 | . . . . . 6 | |
9 | 8 | bibi2d 221 | . . . . 5 |
10 | 7, 9 | imbi12d 223 | . . . 4 |
11 | nfv 1421 | . . . . . 6 | |
12 | dfoprab4f.y | . . . . . . 7 | |
13 | nfs1v 1815 | . . . . . . 7 | |
14 | 12, 13 | nfbi 1481 | . . . . . 6 |
15 | 11, 14 | nfim 1464 | . . . . 5 |
16 | opeq2 3550 | . . . . . . 7 | |
17 | 16 | eqeq2d 2051 | . . . . . 6 |
18 | sbequ12 1654 | . . . . . . 7 | |
19 | 18 | bibi2d 221 | . . . . . 6 |
20 | 17, 19 | imbi12d 223 | . . . . 5 |
21 | dfoprab4f.1 | . . . . 5 | |
22 | 15, 20, 21 | chvar 1640 | . . . 4 |
23 | 5, 10, 22 | chvar 1640 | . . 3 |
24 | 23 | dfoprab4 5818 | . 2 |
25 | nfv 1421 | . . 3 | |
26 | nfv 1421 | . . 3 | |
27 | nfv 1421 | . . . 4 | |
28 | 27, 3 | nfan 1457 | . . 3 |
29 | nfv 1421 | . . . 4 | |
30 | 13 | nfsb 1822 | . . . 4 |
31 | 29, 30 | nfan 1457 | . . 3 |
32 | eleq1 2100 | . . . . 5 | |
33 | eleq1 2100 | . . . . 5 | |
34 | 32, 33 | bi2anan9 538 | . . . 4 |
35 | 18, 8 | sylan9bbr 436 | . . . 4 |
36 | 34, 35 | anbi12d 442 | . . 3 |
37 | 25, 26, 28, 31, 36 | cbvoprab12 5578 | . 2 |
38 | 24, 37 | eqtr4i 2063 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wnf 1349 wcel 1393 wsb 1645 cop 3378 copab 3817 cxp 4343 coprab 5513 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-oprab 5516 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: (None) |
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