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Mirrors > Home > ILE Home > Th. List > prel12 | Unicode version |
Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) |
Ref | Expression |
---|---|
preq12b.1 | |
preq12b.2 | |
preq12b.3 | |
preq12b.4 |
Ref | Expression |
---|---|
prel12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 | . . . . 5 | |
2 | 1 | prid1 3476 | . . . 4 |
3 | eleq2 2101 | . . . 4 | |
4 | 2, 3 | mpbii 136 | . . 3 |
5 | preq12b.2 | . . . . 5 | |
6 | 5 | prid2 3477 | . . . 4 |
7 | eleq2 2101 | . . . 4 | |
8 | 6, 7 | mpbii 136 | . . 3 |
9 | 4, 8 | jca 290 | . 2 |
10 | 1 | elpr 3396 | . . . 4 |
11 | eqeq2 2049 | . . . . . . . . . . . 12 | |
12 | 11 | notbid 592 | . . . . . . . . . . 11 |
13 | orel2 645 | . . . . . . . . . . 11 | |
14 | 12, 13 | syl6bi 152 | . . . . . . . . . 10 |
15 | 14 | com3l 75 | . . . . . . . . 9 |
16 | 15 | imp 115 | . . . . . . . 8 |
17 | 16 | ancrd 309 | . . . . . . 7 |
18 | eqeq2 2049 | . . . . . . . . . . . 12 | |
19 | 18 | notbid 592 | . . . . . . . . . . 11 |
20 | orel1 644 | . . . . . . . . . . 11 | |
21 | 19, 20 | syl6bi 152 | . . . . . . . . . 10 |
22 | 21 | com3l 75 | . . . . . . . . 9 |
23 | 22 | imp 115 | . . . . . . . 8 |
24 | 23 | ancrd 309 | . . . . . . 7 |
25 | 17, 24 | orim12d 700 | . . . . . 6 |
26 | 5 | elpr 3396 | . . . . . . 7 |
27 | orcom 647 | . . . . . . 7 | |
28 | 26, 27 | bitri 173 | . . . . . 6 |
29 | preq12b.3 | . . . . . . 7 | |
30 | preq12b.4 | . . . . . . 7 | |
31 | 1, 5, 29, 30 | preq12b 3541 | . . . . . 6 |
32 | 25, 28, 31 | 3imtr4g 194 | . . . . 5 |
33 | 32 | ex 108 | . . . 4 |
34 | 10, 33 | syl5bi 141 | . . 3 |
35 | 34 | impd 242 | . 2 |
36 | 9, 35 | impbid2 131 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 cvv 2557 cpr 3376 |
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-in1 544 ax-in2 545 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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 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-sn 3381 df-pr 3382 |
This theorem is referenced by: (None) |
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