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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 2912 | . 2 | |
2 | csbeq1 3006 | . . 3 | |
3 | csbeq1 3006 | . . 3 | |
4 | csbeq1 3006 | . . 3 | |
5 | 2, 3, 4 | breq123d 3943 | . 2 |
6 | nfcsb1v 3035 | . . . 4 | |
7 | nfcsb1v 3035 | . . . 4 | |
8 | nfcsb1v 3035 | . . . 4 | |
9 | 6, 7, 8 | nfbr 3974 | . . 3 |
10 | csbeq1a 3012 | . . . 4 | |
11 | csbeq1a 3012 | . . . 4 | |
12 | csbeq1a 3012 | . . . 4 | |
13 | 10, 11, 12 | breq123d 3943 | . . 3 |
14 | 9, 13 | sbie 1764 | . 2 |
15 | 1, 5, 14 | vtoclbg 2747 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wcel 1480 wsb 1735 wsbc 2909 csb 3003 class class class wbr 3929 |
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 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 |
This theorem is referenced by: sbcbr12g 3983 csbcnvg 4723 sbcfung 5147 csbfv12g 5457 |
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