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Mirrors > Home > ILE Home > Th. List > elabgt | Unicode version |
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2688.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
elabgt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2028 | . . . . . . 7 | |
2 | eleq1 2100 | . . . . . . 7 | |
3 | 1, 2 | syl5bbr 183 | . . . . . 6 |
4 | 3 | bibi1d 222 | . . . . 5 |
5 | 4 | biimpd 132 | . . . 4 |
6 | 5 | a2i 11 | . . 3 |
7 | 6 | alimi 1344 | . 2 |
8 | nfcv 2178 | . . . 4 | |
9 | nfab1 2180 | . . . . . 6 | |
10 | 9 | nfel2 2190 | . . . . 5 |
11 | nfv 1421 | . . . . 5 | |
12 | 10, 11 | nfbi 1481 | . . . 4 |
13 | pm5.5 231 | . . . 4 | |
14 | 8, 12, 13 | spcgf 2635 | . . 3 |
15 | 14 | imp 115 | . 2 |
16 | 7, 15 | sylan2 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 cab 2026 |
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-v 2559 |
This theorem is referenced by: elrab3t 2697 |
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