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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 2841.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
elrab3t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . 3 | |
2 | nfa1 1521 | . . . . 5 | |
3 | nfv 1508 | . . . . 5 | |
4 | 2, 3 | nfan 1544 | . . . 4 |
5 | simpl 108 | . . . . . 6 | |
6 | 5 | 19.21bi 1537 | . . . . 5 |
7 | eleq1 2202 | . . . . . . . . . 10 | |
8 | 7 | biimparc 297 | . . . . . . . . 9 |
9 | 8 | biantrurd 303 | . . . . . . . 8 |
10 | 9 | bibi1d 232 | . . . . . . 7 |
11 | 10 | pm5.74da 439 | . . . . . 6 |
12 | 11 | adantl 275 | . . . . 5 |
13 | 6, 12 | mpbid 146 | . . . 4 |
14 | 4, 13 | alrimi 1502 | . . 3 |
15 | elabgt 2825 | . . 3 | |
16 | 1, 14, 15 | syl2anc 408 | . 2 |
17 | df-rab 2425 | . . . 4 | |
18 | 17 | eleq2i 2206 | . . 3 |
19 | 18 | bibi1i 227 | . 2 |
20 | 16, 19 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 cab 2125 crab 2420 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 |
This theorem is referenced by: f1oresrab 5585 |
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