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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 1508 | . . . . 5 | |
2 | dfoprab4f.x | . . . . . 6 | |
3 | nfs1v 1912 | . . . . . 6 | |
4 | 2, 3 | nfbi 1568 | . . . . 5 |
5 | 1, 4 | nfim 1551 | . . . 4 |
6 | opeq1 3705 | . . . . . 6 | |
7 | 6 | eqeq2d 2151 | . . . . 5 |
8 | sbequ12 1744 | . . . . . 6 | |
9 | 8 | bibi2d 231 | . . . . 5 |
10 | 7, 9 | imbi12d 233 | . . . 4 |
11 | nfv 1508 | . . . . . 6 | |
12 | dfoprab4f.y | . . . . . . 7 | |
13 | nfs1v 1912 | . . . . . . 7 | |
14 | 12, 13 | nfbi 1568 | . . . . . 6 |
15 | 11, 14 | nfim 1551 | . . . . 5 |
16 | opeq2 3706 | . . . . . . 7 | |
17 | 16 | eqeq2d 2151 | . . . . . 6 |
18 | sbequ12 1744 | . . . . . . 7 | |
19 | 18 | bibi2d 231 | . . . . . 6 |
20 | 17, 19 | imbi12d 233 | . . . . 5 |
21 | dfoprab4f.1 | . . . . 5 | |
22 | 15, 20, 21 | chvar 1730 | . . . 4 |
23 | 5, 10, 22 | chvar 1730 | . . 3 |
24 | 23 | dfoprab4 6090 | . 2 |
25 | nfv 1508 | . . 3 | |
26 | nfv 1508 | . . 3 | |
27 | nfv 1508 | . . . 4 | |
28 | 27, 3 | nfan 1544 | . . 3 |
29 | nfv 1508 | . . . 4 | |
30 | 13 | nfsb 1919 | . . . 4 |
31 | 29, 30 | nfan 1544 | . . 3 |
32 | eleq1 2202 | . . . . 5 | |
33 | eleq1 2202 | . . . . 5 | |
34 | 32, 33 | bi2anan9 595 | . . . 4 |
35 | 18, 8 | sylan9bbr 458 | . . . 4 |
36 | 34, 35 | anbi12d 464 | . . 3 |
37 | 25, 26, 28, 31, 36 | cbvoprab12 5845 | . 2 |
38 | 24, 37 | eqtr4i 2163 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wnf 1436 wcel 1480 wsb 1735 cop 3530 copab 3988 cxp 4537 coprab 5775 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-oprab 5778 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: (None) |
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