Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcel12g Unicode version

Theorem sbcel12g 3046
 Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g

Proof of Theorem sbcel12g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2940 . . 3
2 dfsbcq2 2940 . . . . 5
32abbidv 2275 . . . 4
4 dfsbcq2 2940 . . . . 5
54abbidv 2275 . . . 4
63, 5eleq12d 2228 . . 3
7 nfs1v 1919 . . . . . 6
87nfab 2304 . . . . 5
9 nfs1v 1919 . . . . . 6
109nfab 2304 . . . . 5
118, 10nfel 2308 . . . 4
12 sbab 2285 . . . . 5
13 sbab 2285 . . . . 5
1412, 13eleq12d 2228 . . . 4
1511, 14sbie 1771 . . 3
161, 6, 15vtoclbg 2773 . 2
17 df-csb 3032 . . 3
18 df-csb 3032 . . 3
1917, 18eleq12i 2225 . 2
2016, 19bitr4di 197 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1335  wsb 1742   wcel 2128  cab 2143  wsbc 2937  csb 3031 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938  df-csb 3032 This theorem is referenced by:  sbcnel12g  3048  sbcel1g  3050  sbcel2g  3052  sbccsb2g  3061  ixpsnval  6639
 Copyright terms: Public domain W3C validator