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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 3665. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
preqr1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 3597 | . . . . . . 7 | |
2 | eleq2 2181 | . . . . . . 7 | |
3 | 1, 2 | syl5ibcom 154 | . . . . . 6 |
4 | elprg 3517 | . . . . . 6 | |
5 | 3, 4 | sylibd 148 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | 6 | imp 123 | . . 3 |
8 | prid1g 3597 | . . . . . . 7 | |
9 | eleq2 2181 | . . . . . . 7 | |
10 | 8, 9 | syl5ibrcom 156 | . . . . . 6 |
11 | elprg 3517 | . . . . . 6 | |
12 | 10, 11 | sylibd 148 | . . . . 5 |
13 | 12 | adantl 275 | . . . 4 |
14 | 13 | imp 123 | . . 3 |
15 | eqcom 2119 | . . 3 | |
16 | eqeq2 2127 | . . 3 | |
17 | 7, 14, 15, 16 | oplem1 944 | . 2 |
18 | 17 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wceq 1316 wcel 1465 cvv 2660 cpr 3498 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 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-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: preqr2g 3664 |
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