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Mirrors > Home > ILE Home > Th. List > csbiebt | Unicode version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3043.) (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
csbiebt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 | |
2 | spsbc 2920 | . . . . 5 | |
3 | 2 | adantr 274 | . . . 4 |
4 | simpl 108 | . . . . 5 | |
5 | biimt 240 | . . . . . . 7 | |
6 | csbeq1a 3012 | . . . . . . . 8 | |
7 | 6 | eqeq1d 2148 | . . . . . . 7 |
8 | 5, 7 | bitr3d 189 | . . . . . 6 |
9 | 8 | adantl 275 | . . . . 5 |
10 | nfv 1508 | . . . . . 6 | |
11 | nfnfc1 2284 | . . . . . 6 | |
12 | 10, 11 | nfan 1544 | . . . . 5 |
13 | nfcsb1v 3035 | . . . . . . 7 | |
14 | 13 | a1i 9 | . . . . . 6 |
15 | simpr 109 | . . . . . 6 | |
16 | 14, 15 | nfeqd 2296 | . . . . 5 |
17 | 4, 9, 12, 16 | sbciedf 2944 | . . . 4 |
18 | 3, 17 | sylibd 148 | . . 3 |
19 | 13 | a1i 9 | . . . . . . . 8 |
20 | id 19 | . . . . . . . 8 | |
21 | 19, 20 | nfeqd 2296 | . . . . . . 7 |
22 | 11, 21 | nfan1 1543 | . . . . . 6 |
23 | 7 | biimprcd 159 | . . . . . . 7 |
24 | 23 | adantl 275 | . . . . . 6 |
25 | 22, 24 | alrimi 1502 | . . . . 5 |
26 | 25 | ex 114 | . . . 4 |
27 | 26 | adantl 275 | . . 3 |
28 | 18, 27 | impbid 128 | . 2 |
29 | 1, 28 | sylan 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 wnfc 2268 cvv 2686 wsbc 2909 csb 3003 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 df-csb 3004 |
This theorem is referenced by: csbiedf 3040 csbieb 3041 csbiegf 3043 |
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