Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > opabex3d | Unicode version |
Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) |
Ref | Expression |
---|---|
opabex3d.1 | |
opabex3d.2 |
Ref | Expression |
---|---|
opabex3d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42v 1827 | . . . . . 6 | |
2 | an12 525 | . . . . . . 7 | |
3 | 2 | exbii 1536 | . . . . . 6 |
4 | elxp 4380 | . . . . . . . 8 | |
5 | excom 1594 | . . . . . . . . 9 | |
6 | an12 525 | . . . . . . . . . . . . 13 | |
7 | velsn 3415 | . . . . . . . . . . . . . 14 | |
8 | 7 | anbi1i 445 | . . . . . . . . . . . . 13 |
9 | 6, 8 | bitri 182 | . . . . . . . . . . . 12 |
10 | 9 | exbii 1536 | . . . . . . . . . . 11 |
11 | vex 2604 | . . . . . . . . . . . 12 | |
12 | opeq1 3570 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqeq2d 2092 | . . . . . . . . . . . . 13 |
14 | 13 | anbi1d 452 | . . . . . . . . . . . 12 |
15 | 11, 14 | ceqsexv 2638 | . . . . . . . . . . 11 |
16 | 10, 15 | bitri 182 | . . . . . . . . . 10 |
17 | 16 | exbii 1536 | . . . . . . . . 9 |
18 | 5, 17 | bitri 182 | . . . . . . . 8 |
19 | nfv 1461 | . . . . . . . . . 10 | |
20 | nfsab1 2071 | . . . . . . . . . 10 | |
21 | 19, 20 | nfan 1497 | . . . . . . . . 9 |
22 | nfv 1461 | . . . . . . . . 9 | |
23 | opeq2 3571 | . . . . . . . . . . 11 | |
24 | 23 | eqeq2d 2092 | . . . . . . . . . 10 |
25 | sbequ12 1694 | . . . . . . . . . . . 12 | |
26 | 25 | equcoms 1634 | . . . . . . . . . . 11 |
27 | df-clab 2068 | . . . . . . . . . . 11 | |
28 | 26, 27 | syl6rbbr 197 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 456 | . . . . . . . . 9 |
30 | 21, 22, 29 | cbvex 1679 | . . . . . . . 8 |
31 | 4, 18, 30 | 3bitri 204 | . . . . . . 7 |
32 | 31 | anbi2i 444 | . . . . . 6 |
33 | 1, 3, 32 | 3bitr4ri 211 | . . . . 5 |
34 | 33 | exbii 1536 | . . . 4 |
35 | eliun 3682 | . . . . 5 | |
36 | df-rex 2354 | . . . . 5 | |
37 | 35, 36 | bitri 182 | . . . 4 |
38 | elopab 4013 | . . . 4 | |
39 | 34, 37, 38 | 3bitr4i 210 | . . 3 |
40 | 39 | eqriv 2078 | . 2 |
41 | opabex3d.1 | . . 3 | |
42 | snexg 3956 | . . . . . 6 | |
43 | 11, 42 | ax-mp 7 | . . . . 5 |
44 | opabex3d.2 | . . . . 5 | |
45 | xpexg 4470 | . . . . 5 | |
46 | 43, 44, 45 | sylancr 405 | . . . 4 |
47 | 46 | ralrimiva 2434 | . . 3 |
48 | iunexg 5766 | . . 3 | |
49 | 41, 47, 48 | syl2anc 403 | . 2 |
50 | 40, 49 | syl5eqelr 2166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wsb 1685 cab 2067 wral 2348 wrex 2349 cvv 2601 csn 3398 cop 3401 ciun 3678 copab 3838 cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: ovshftex 9707 |
Copyright terms: Public domain | W3C validator |