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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 4223 | . . . . . . 7 |
4 | 3 | anbi2i 454 | . . . . . 6 |
5 | ancom 264 | . . . . . 6 | |
6 | anass 399 | . . . . . 6 | |
7 | 4, 5, 6 | 3bitri 205 | . . . . 5 |
8 | 7 | exbii 1598 | . . . 4 |
9 | 19.42v 1899 | . . . 4 | |
10 | opeq2 3764 | . . . . . . 7 | |
11 | 10 | eqeq2d 2182 | . . . . . 6 |
12 | 2, 11 | ceqsexv 2769 | . . . . 5 |
13 | 12 | anbi2i 454 | . . . 4 |
14 | 8, 9, 13 | 3bitri 205 | . . 3 |
15 | 14 | exbii 1598 | . 2 |
16 | opeq1 3763 | . . . 4 | |
17 | 16 | eqeq2d 2182 | . . 3 |
18 | 1, 17 | ceqsexv 2769 | . 2 |
19 | 15, 18 | bitr2i 184 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cvv 2730 cop 3584 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 |
This theorem is referenced by: copsexg 4227 ralxpf 4755 rexxpf 4756 oprabid 5882 |
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