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Mirrors > Home > MPE Home > Th. List > csbiebt | Unicode version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3458.) (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
csbiebt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3118 | . 2 | |
2 | spsbc 3340 | . . . . 5 | |
3 | 2 | adantr 465 | . . . 4 |
4 | simpl 457 | . . . . 5 | |
5 | biimt 335 | . . . . . . 7 | |
6 | csbeq1a 3443 | . . . . . . . 8 | |
7 | 6 | eqeq1d 2459 | . . . . . . 7 |
8 | 5, 7 | bitr3d 255 | . . . . . 6 |
9 | 8 | adantl 466 | . . . . 5 |
10 | nfv 1707 | . . . . . 6 | |
11 | nfnfc1 2622 | . . . . . 6 | |
12 | 10, 11 | nfan 1928 | . . . . 5 |
13 | nfcsb1v 3450 | . . . . . . 7 | |
14 | 13 | a1i 11 | . . . . . 6 |
15 | simpr 461 | . . . . . 6 | |
16 | 14, 15 | nfeqd 2626 | . . . . 5 |
17 | 4, 9, 12, 16 | sbciedf 3363 | . . . 4 |
18 | 3, 17 | sylibd 214 | . . 3 |
19 | 13 | a1i 11 | . . . . . . . 8 |
20 | id 22 | . . . . . . . 8 | |
21 | 19, 20 | nfeqd 2626 | . . . . . . 7 |
22 | 11, 21 | nfan1 1927 | . . . . . 6 |
23 | 7 | biimprcd 225 | . . . . . . 7 |
24 | 23 | adantl 466 | . . . . . 6 |
25 | 22, 24 | alrimi 1877 | . . . . 5 |
26 | 25 | ex 434 | . . . 4 |
27 | 26 | adantl 466 | . . 3 |
28 | 18, 27 | impbid 191 | . 2 |
29 | 1, 28 | sylan 471 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 A. wal 1393 = wceq 1395
e. wcel 1818 F/_ wnfc 2605 cvv 3109
[. wsbc 3327 [_ csb 3434 |
This theorem is referenced by: csbiedf 3455 csbieb 3456 csbiegf 3458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 |
This theorem depends on definitions: df-bi 185 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-v 3111 df-sbc 3328 df-csb 3435 |
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