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Mirrors > Home > ILE Home > Th. List > cleqh | Unicode version |
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2282. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cleqh.1 | |
cleqh.2 |
Ref | Expression |
---|---|
cleqh |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2111 | . 2 | |
2 | ax-17 1491 | . . . 4 | |
3 | dfbi2 385 | . . . . 5 | |
4 | cleqh.1 | . . . . . . 7 | |
5 | cleqh.2 | . . . . . . 7 | |
6 | 4, 5 | hbim 1509 | . . . . . 6 |
7 | 5, 4 | hbim 1509 | . . . . . 6 |
8 | 6, 7 | hban 1511 | . . . . 5 |
9 | 3, 8 | hbxfrbi 1433 | . . . 4 |
10 | eleq1 2180 | . . . . . 6 | |
11 | eleq1 2180 | . . . . . 6 | |
12 | 10, 11 | bibi12d 234 | . . . . 5 |
13 | 12 | biimpd 143 | . . . 4 |
14 | 2, 9, 13 | cbv3h 1706 | . . 3 |
15 | 12 | equcoms 1669 | . . . . 5 |
16 | 15 | biimprd 157 | . . . 4 |
17 | 9, 2, 16 | cbv3h 1706 | . . 3 |
18 | 14, 17 | impbii 125 | . 2 |
19 | 1, 18 | bitr4i 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wcel 1465 |
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-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: abeq2 2226 |
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