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Mirrors > Home > ILE Home > Th. List > sbcssg | Unicode version |
Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
Ref | Expression |
---|---|
sbcssg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcalg 2811 | . . 3 | |
2 | sbcimg 2804 | . . . . 5 | |
3 | sbcel2g 2871 | . . . . . 6 | |
4 | sbcel2g 2871 | . . . . . 6 | |
5 | 3, 4 | imbi12d 223 | . . . . 5 |
6 | 2, 5 | bitrd 177 | . . . 4 |
7 | 6 | albidv 1705 | . . 3 |
8 | 1, 7 | bitrd 177 | . 2 |
9 | dfss2 2934 | . . 3 | |
10 | 9 | sbcbii 2818 | . 2 |
11 | dfss2 2934 | . 2 | |
12 | 8, 10, 11 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wal 1241 wcel 1393 wsbc 2764 csb 2852 wss 2917 |
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-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-in 2924 df-ss 2931 |
This theorem is referenced by: sbcrel 4426 sbcfg 5045 |
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