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Mirrors > Home > ILE Home > Th. List > eqrd | Unicode version |
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
Ref | Expression |
---|---|
eqrd.0 | |
eqrd.1 | |
eqrd.2 | |
eqrd.3 |
Ref | Expression |
---|---|
eqrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrd.0 | . . 3 | |
2 | eqrd.1 | . . 3 | |
3 | eqrd.2 | . . 3 | |
4 | eqrd.3 | . . . 4 | |
5 | 4 | biimpd 142 | . . 3 |
6 | 1, 2, 3, 5 | ssrd 3004 | . 2 |
7 | 4 | biimprd 156 | . . 3 |
8 | 1, 3, 2, 7 | ssrd 3004 | . 2 |
9 | 6, 8 | eqssd 3016 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wceq 1284 wnf 1389 wcel 1433 wnfc 2206 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-in 2979 df-ss 2986 |
This theorem is referenced by: (None) |
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