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Mirrors > Home > ILE Home > Th. List > preqr1g | Unicode version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3560. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
preqr1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 3496 | . . . . . . 7 | |
2 | eleq2 2142 | . . . . . . 7 | |
3 | 1, 2 | syl5ibcom 153 | . . . . . 6 |
4 | elprg 3418 | . . . . . 6 | |
5 | 3, 4 | sylibd 147 | . . . . 5 |
6 | 5 | adantr 270 | . . . 4 |
7 | 6 | imp 122 | . . 3 |
8 | prid1g 3496 | . . . . . . 7 | |
9 | eleq2 2142 | . . . . . . 7 | |
10 | 8, 9 | syl5ibrcom 155 | . . . . . 6 |
11 | elprg 3418 | . . . . . 6 | |
12 | 10, 11 | sylibd 147 | . . . . 5 |
13 | 12 | adantl 271 | . . . 4 |
14 | 13 | imp 122 | . . 3 |
15 | eqcom 2083 | . . 3 | |
16 | eqeq2 2090 | . . 3 | |
17 | 7, 14, 15, 16 | oplem1 916 | . 2 |
18 | 17 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wceq 1284 wcel 1433 cvv 2601 cpr 3399 |
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-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: preqr2g 3559 |
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