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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 143 | . . 3 |
6 | 1, 2, 3, 5 | ssrd 3102 | . 2 |
7 | 4 | biimprd 157 | . . 3 |
8 | 1, 3, 2, 7 | ssrd 3102 | . 2 |
9 | 6, 8 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wnf 1436 wcel 1480 wnfc 2268 |
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-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-in 3077 df-ss 3084 |
This theorem is referenced by: dfss4st 3309 imasnopn 12468 |
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