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Mirrors > Home > ILE Home > Th. List > sbcbrg | Unicode version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
sbcbrg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2767 | . 2 | |
2 | csbeq1 2855 | . . 3 | |
3 | csbeq1 2855 | . . 3 | |
4 | csbeq1 2855 | . . 3 | |
5 | 2, 3, 4 | breq123d 3778 | . 2 |
6 | nfcsb1v 2882 | . . . 4 | |
7 | nfcsb1v 2882 | . . . 4 | |
8 | nfcsb1v 2882 | . . . 4 | |
9 | 6, 7, 8 | nfbr 3808 | . . 3 |
10 | csbeq1a 2860 | . . . 4 | |
11 | csbeq1a 2860 | . . . 4 | |
12 | csbeq1a 2860 | . . . 4 | |
13 | 10, 11, 12 | breq123d 3778 | . . 3 |
14 | 9, 13 | sbie 1674 | . 2 |
15 | 1, 5, 14 | vtoclbg 2614 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 wcel 1393 wsb 1645 wsbc 2764 csb 2852 class class class wbr 3764 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 |
This theorem is referenced by: sbcbr12g 3814 csbcnvg 4519 sbcfung 4925 csbfv12g 5209 |
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