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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 3687 | . . . 4 |
3 | eleq2 2234 | . . . 4 | |
4 | 2, 3 | mpbii 147 | . . 3 |
5 | preq12b.2 | . . . . 5 | |
6 | 5 | prid2 3688 | . . . 4 |
7 | eleq2 2234 | . . . 4 | |
8 | 6, 7 | mpbii 147 | . . 3 |
9 | 4, 8 | jca 304 | . 2 |
10 | 1 | elpr 3602 | . . . 4 |
11 | eqeq2 2180 | . . . . . . . . . . . 12 | |
12 | 11 | notbid 662 | . . . . . . . . . . 11 |
13 | orel2 721 | . . . . . . . . . . 11 | |
14 | 12, 13 | syl6bi 162 | . . . . . . . . . 10 |
15 | 14 | com3l 81 | . . . . . . . . 9 |
16 | 15 | imp 123 | . . . . . . . 8 |
17 | 16 | ancrd 324 | . . . . . . 7 |
18 | eqeq2 2180 | . . . . . . . . . . . 12 | |
19 | 18 | notbid 662 | . . . . . . . . . . 11 |
20 | orel1 720 | . . . . . . . . . . 11 | |
21 | 19, 20 | syl6bi 162 | . . . . . . . . . 10 |
22 | 21 | com3l 81 | . . . . . . . . 9 |
23 | 22 | imp 123 | . . . . . . . 8 |
24 | 23 | ancrd 324 | . . . . . . 7 |
25 | 17, 24 | orim12d 781 | . . . . . 6 |
26 | 5 | elpr 3602 | . . . . . . 7 |
27 | orcom 723 | . . . . . . 7 | |
28 | 26, 27 | bitri 183 | . . . . . 6 |
29 | preq12b.3 | . . . . . . 7 | |
30 | preq12b.4 | . . . . . . 7 | |
31 | 1, 5, 29, 30 | preq12b 3755 | . . . . . 6 |
32 | 25, 28, 31 | 3imtr4g 204 | . . . . 5 |
33 | 32 | ex 114 | . . . 4 |
34 | 10, 33 | syl5bi 151 | . . 3 |
35 | 34 | impd 252 | . 2 |
36 | 9, 35 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 cvv 2730 cpr 3582 |
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-in1 609 ax-in2 610 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-ext 2152 |
This theorem depends on definitions: df-bi 116 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-sn 3587 df-pr 3588 |
This theorem is referenced by: (None) |
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