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Mirrors > Home > NFE Home > Th. List > sbcel12g | Unicode version |
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbcel12g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3049 | . . 3 | |
2 | dfsbcq2 3049 | . . . . 5 | |
3 | 2 | abbidv 2467 | . . . 4 |
4 | dfsbcq2 3049 | . . . . 5 | |
5 | 4 | abbidv 2467 | . . . 4 |
6 | 3, 5 | eleq12d 2421 | . . 3 |
7 | nfs1v 2106 | . . . . . 6 | |
8 | 7 | nfab 2493 | . . . . 5 |
9 | nfs1v 2106 | . . . . . 6 | |
10 | 9 | nfab 2493 | . . . . 5 |
11 | 8, 10 | nfel 2497 | . . . 4 |
12 | sbab 2475 | . . . . 5 | |
13 | sbab 2475 | . . . . 5 | |
14 | 12, 13 | eleq12d 2421 | . . . 4 |
15 | 11, 14 | sbie 2038 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2915 | . 2 |
17 | df-csb 3137 | . . 3 | |
18 | df-csb 3137 | . . 3 | |
19 | 17, 18 | eleq12i 2418 | . 2 |
20 | 16, 19 | syl6bbr 254 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wceq 1642 wsb 1648 wcel 1710 cab 2339 wsbc 3046 csb 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: sbcnel12g 3153 sbcel1g 3155 sbcel2g 3157 sbccsb2g 3165 |
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