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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 3703. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
preqr1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 3635 | . . . . . . 7 | |
2 | eleq2 2204 | . . . . . . 7 | |
3 | 1, 2 | syl5ibcom 154 | . . . . . 6 |
4 | elprg 3552 | . . . . . 6 | |
5 | 3, 4 | sylibd 148 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | 6 | imp 123 | . . 3 |
8 | prid1g 3635 | . . . . . . 7 | |
9 | eleq2 2204 | . . . . . . 7 | |
10 | 8, 9 | syl5ibrcom 156 | . . . . . 6 |
11 | elprg 3552 | . . . . . 6 | |
12 | 10, 11 | sylibd 148 | . . . . 5 |
13 | 12 | adantl 275 | . . . 4 |
14 | 13 | imp 123 | . . 3 |
15 | eqcom 2142 | . . 3 | |
16 | eqeq2 2150 | . . 3 | |
17 | 7, 14, 15, 16 | oplem1 960 | . 2 |
18 | 17 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1332 wcel 1481 cvv 2689 cpr 3533 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: preqr2g 3702 |
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