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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 2933 | . . 3 | |
2 | sbcimg 2922 | . . . . 5 | |
3 | sbcel2g 2994 | . . . . . 6 | |
4 | sbcel2g 2994 | . . . . . 6 | |
5 | 3, 4 | imbi12d 233 | . . . . 5 |
6 | 2, 5 | bitrd 187 | . . . 4 |
7 | 6 | albidv 1780 | . . 3 |
8 | 1, 7 | bitrd 187 | . 2 |
9 | dfss2 3056 | . . 3 | |
10 | 9 | sbcbii 2940 | . 2 |
11 | dfss2 3056 | . 2 | |
12 | 8, 10, 11 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1314 wcel 1465 wsbc 2882 csb 2975 wss 3041 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sbc 2883 df-csb 2976 df-in 3047 df-ss 3054 |
This theorem is referenced by: sbcrel 4595 sbcfg 5241 |
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