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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 1515 | . . . . 5 | |
2 | dfoprab4f.x | . . . . . 6 | |
3 | nfs1v 1926 | . . . . . 6 | |
4 | 2, 3 | nfbi 1576 | . . . . 5 |
5 | 1, 4 | nfim 1559 | . . . 4 |
6 | opeq1 3752 | . . . . . 6 | |
7 | 6 | eqeq2d 2176 | . . . . 5 |
8 | sbequ12 1758 | . . . . . 6 | |
9 | 8 | bibi2d 231 | . . . . 5 |
10 | 7, 9 | imbi12d 233 | . . . 4 |
11 | nfv 1515 | . . . . . 6 | |
12 | dfoprab4f.y | . . . . . . 7 | |
13 | nfs1v 1926 | . . . . . . 7 | |
14 | 12, 13 | nfbi 1576 | . . . . . 6 |
15 | 11, 14 | nfim 1559 | . . . . 5 |
16 | opeq2 3753 | . . . . . . 7 | |
17 | 16 | eqeq2d 2176 | . . . . . 6 |
18 | sbequ12 1758 | . . . . . . 7 | |
19 | 18 | bibi2d 231 | . . . . . 6 |
20 | 17, 19 | imbi12d 233 | . . . . 5 |
21 | dfoprab4f.1 | . . . . 5 | |
22 | 15, 20, 21 | chvar 1744 | . . . 4 |
23 | 5, 10, 22 | chvar 1744 | . . 3 |
24 | 23 | dfoprab4 6152 | . 2 |
25 | nfv 1515 | . . 3 | |
26 | nfv 1515 | . . 3 | |
27 | nfv 1515 | . . . 4 | |
28 | 27, 3 | nfan 1552 | . . 3 |
29 | nfv 1515 | . . . 4 | |
30 | 13 | nfsb 1933 | . . . 4 |
31 | 29, 30 | nfan 1552 | . . 3 |
32 | eleq1 2227 | . . . . 5 | |
33 | eleq1 2227 | . . . . 5 | |
34 | 32, 33 | bi2anan9 596 | . . . 4 |
35 | 18, 8 | sylan9bbr 459 | . . . 4 |
36 | 34, 35 | anbi12d 465 | . . 3 |
37 | 25, 26, 28, 31, 36 | cbvoprab12 5907 | . 2 |
38 | 24, 37 | eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wnf 1447 wsb 1749 wcel 2135 cop 3573 copab 4036 cxp 4596 coprab 5837 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-oprab 5840 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: (None) |
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