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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 2879.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
elrab3t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . 3 | |
2 | nfa1 1528 | . . . . 5 | |
3 | nfv 1515 | . . . . 5 | |
4 | 2, 3 | nfan 1552 | . . . 4 |
5 | simpl 108 | . . . . . 6 | |
6 | 5 | 19.21bi 1545 | . . . . 5 |
7 | eleq1 2227 | . . . . . . . . . 10 | |
8 | 7 | biimparc 297 | . . . . . . . . 9 |
9 | 8 | biantrurd 303 | . . . . . . . 8 |
10 | 9 | bibi1d 232 | . . . . . . 7 |
11 | 10 | pm5.74da 440 | . . . . . 6 |
12 | 11 | adantl 275 | . . . . 5 |
13 | 6, 12 | mpbid 146 | . . . 4 |
14 | 4, 13 | alrimi 1509 | . . 3 |
15 | elabgt 2863 | . . 3 | |
16 | 1, 14, 15 | syl2anc 409 | . 2 |
17 | df-rab 2451 | . . . 4 | |
18 | 17 | eleq2i 2231 | . . 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 1340 wceq 1342 wcel 2135 cab 2150 crab 2446 |
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-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rab 2451 df-v 2724 |
This theorem is referenced by: f1oresrab 5645 |
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