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Mirrors > Home > ILE Home > Th. List > cnveqb | Unicode version |
Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
cnveqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 4772 | . 2 | |
2 | dfrel2 5048 | . . . 4 | |
3 | dfrel2 5048 | . . . . . . 7 | |
4 | cnveq 4772 | . . . . . . . . 9 | |
5 | eqeq2 2174 | . . . . . . . . 9 | |
6 | 4, 5 | syl5ibr 155 | . . . . . . . 8 |
7 | 6 | eqcoms 2167 | . . . . . . 7 |
8 | 3, 7 | sylbi 120 | . . . . . 6 |
9 | eqeq1 2171 | . . . . . . 7 | |
10 | 9 | imbi2d 229 | . . . . . 6 |
11 | 8, 10 | syl5ibr 155 | . . . . 5 |
12 | 11 | eqcoms 2167 | . . . 4 |
13 | 2, 12 | sylbi 120 | . . 3 |
14 | 13 | imp 123 | . 2 |
15 | 1, 14 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 ccnv 4597 wrel 4603 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 |
This theorem is referenced by: cnveq0 5054 |
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