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Mirrors > Home > ILE Home > Th. List > elrab3t | Unicode version |
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2699.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
elrab3t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . 3 | |
2 | nfa1 1434 | . . . . 5 | |
3 | nfv 1421 | . . . . 5 | |
4 | 2, 3 | nfan 1457 | . . . 4 |
5 | simpl 102 | . . . . . 6 | |
6 | 5 | 19.21bi 1450 | . . . . 5 |
7 | eleq1 2100 | . . . . . . . . . 10 | |
8 | 7 | biimparc 283 | . . . . . . . . 9 |
9 | 8 | biantrurd 289 | . . . . . . . 8 |
10 | 9 | bibi1d 222 | . . . . . . 7 |
11 | 10 | pm5.74da 417 | . . . . . 6 |
12 | 11 | adantl 262 | . . . . 5 |
13 | 6, 12 | mpbid 135 | . . . 4 |
14 | 4, 13 | alrimi 1415 | . . 3 |
15 | elabgt 2684 | . . 3 | |
16 | 1, 14, 15 | syl2anc 391 | . 2 |
17 | df-rab 2315 | . . . 4 | |
18 | 17 | eleq2i 2104 | . . 3 |
19 | 18 | bibi1i 217 | . 2 |
20 | 16, 19 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 cab 2026 crab 2310 |
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-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 |
This theorem is referenced by: f1oresrab 5329 |
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