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Mirrors > Home > ILE Home > Th. List > eqvinop | Unicode version |
Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
eqvinop.1 | |
eqvinop.2 |
Ref | Expression |
---|---|
eqvinop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvinop.1 | . . . . . . . 8 | |
2 | eqvinop.2 | . . . . . . . 8 | |
3 | 1, 2 | opth2 3977 | . . . . . . 7 |
4 | 3 | anbi2i 430 | . . . . . 6 |
5 | ancom 253 | . . . . . 6 | |
6 | anass 381 | . . . . . 6 | |
7 | 4, 5, 6 | 3bitri 195 | . . . . 5 |
8 | 7 | exbii 1496 | . . . 4 |
9 | 19.42v 1786 | . . . 4 | |
10 | opeq2 3550 | . . . . . . 7 | |
11 | 10 | eqeq2d 2051 | . . . . . 6 |
12 | 2, 11 | ceqsexv 2593 | . . . . 5 |
13 | 12 | anbi2i 430 | . . . 4 |
14 | 8, 9, 13 | 3bitri 195 | . . 3 |
15 | 14 | exbii 1496 | . 2 |
16 | opeq1 3549 | . . . 4 | |
17 | 16 | eqeq2d 2051 | . . 3 |
18 | 1, 17 | ceqsexv 2593 | . 2 |
19 | 15, 18 | bitr2i 174 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: copsexg 3981 ralxpf 4482 rexxpf 4483 oprabid 5537 |
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