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Mirrors > Home > ILE Home > Th. List > opelopabsb | Unicode version |
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
opelopabsb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 3995 | . . . 4 | |
2 | simpl 102 | . . . . . . . 8 | |
3 | 2 | eqcomd 2045 | . . . . . . 7 |
4 | vex 2560 | . . . . . . . 8 | |
5 | vex 2560 | . . . . . . . 8 | |
6 | 4, 5 | opth 3974 | . . . . . . 7 |
7 | 3, 6 | sylib 127 | . . . . . 6 |
8 | 7 | 2eximi 1492 | . . . . 5 |
9 | eeanv 1807 | . . . . . 6 | |
10 | isset 2561 | . . . . . . 7 | |
11 | isset 2561 | . . . . . . 7 | |
12 | 10, 11 | anbi12i 433 | . . . . . 6 |
13 | 9, 12 | bitr4i 176 | . . . . 5 |
14 | 8, 13 | sylib 127 | . . . 4 |
15 | 1, 14 | sylbi 114 | . . 3 |
16 | nfv 1421 | . . . 4 | |
17 | nfv 1421 | . . . 4 | |
18 | nfs1v 1815 | . . . 4 | |
19 | nfs1v 1815 | . . . . 5 | |
20 | 19 | nfsbxy 1818 | . . . 4 |
21 | sbequ12 1654 | . . . . 5 | |
22 | sbequ12 1654 | . . . . 5 | |
23 | 21, 22 | sylan9bbr 436 | . . . 4 |
24 | 16, 17, 18, 20, 23 | cbvopab 3828 | . . 3 |
25 | 15, 24 | eleq2s 2132 | . 2 |
26 | sbcex 2772 | . . 3 | |
27 | spesbc 2843 | . . . 4 | |
28 | sbcex 2772 | . . . . 5 | |
29 | 28 | exlimiv 1489 | . . . 4 |
30 | 27, 29 | syl 14 | . . 3 |
31 | 26, 30 | jca 290 | . 2 |
32 | opeq1 3549 | . . . . 5 | |
33 | 32 | eleq1d 2106 | . . . 4 |
34 | dfsbcq2 2767 | . . . 4 | |
35 | 33, 34 | bibi12d 224 | . . 3 |
36 | opeq2 3550 | . . . . 5 | |
37 | 36 | eleq1d 2106 | . . . 4 |
38 | dfsbcq2 2767 | . . . . 5 | |
39 | 38 | sbcbidv 2817 | . . . 4 |
40 | 37, 39 | bibi12d 224 | . . 3 |
41 | nfopab1 3826 | . . . . . 6 | |
42 | 41 | nfel2 2190 | . . . . 5 |
43 | nfs1v 1815 | . . . . 5 | |
44 | 42, 43 | nfbi 1481 | . . . 4 |
45 | opeq1 3549 | . . . . . 6 | |
46 | 45 | eleq1d 2106 | . . . . 5 |
47 | sbequ12 1654 | . . . . 5 | |
48 | 46, 47 | bibi12d 224 | . . . 4 |
49 | nfopab2 3827 | . . . . . . 7 | |
50 | 49 | nfel2 2190 | . . . . . 6 |
51 | nfs1v 1815 | . . . . . 6 | |
52 | 50, 51 | nfbi 1481 | . . . . 5 |
53 | opeq2 3550 | . . . . . . 7 | |
54 | 53 | eleq1d 2106 | . . . . . 6 |
55 | sbequ12 1654 | . . . . . 6 | |
56 | 54, 55 | bibi12d 224 | . . . . 5 |
57 | opabid 3994 | . . . . 5 | |
58 | 52, 56, 57 | chvar 1640 | . . . 4 |
59 | 44, 48, 58 | chvar 1640 | . . 3 |
60 | 35, 40, 59 | vtocl2g 2617 | . 2 |
61 | 25, 31, 60 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 wsb 1645 cvv 2557 wsbc 2764 cop 3378 copab 3817 |
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-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 |
This theorem is referenced by: brabsb 3998 opelopabaf 4010 opelopabf 4011 difopab 4469 isarep1 4985 |
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