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Theorem sbcel12g 2922
 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 2819 . . 3
2 dfsbcq2 2819 . . . . 5
32abbidv 2197 . . . 4
4 dfsbcq2 2819 . . . . 5
54abbidv 2197 . . . 4
63, 5eleq12d 2150 . . 3
7 nfs1v 1857 . . . . . 6
87nfab 2224 . . . . 5
9 nfs1v 1857 . . . . . 6
109nfab 2224 . . . . 5
118, 10nfel 2228 . . . 4
12 sbab 2206 . . . . 5
13 sbab 2206 . . . . 5
1412, 13eleq12d 2150 . . . 4
1511, 14sbie 1715 . . 3
161, 6, 15vtoclbg 2660 . 2
17 df-csb 2910 . . 3
18 df-csb 2910 . . 3
1917, 18eleq12i 2147 . 2
2016, 19syl6bbr 196 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 103   wceq 1285   wcel 1434  wsb 1686  cab 2068  wsbc 2816  csb 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910 This theorem is referenced by:  sbcnel12g  2924  sbcel1g  2926  sbcel2g  2928  sbccsb2g  2936
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