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Theorem sbcel12g 3266
 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 3164 . . 3
2 dfsbcq2 3164 . . . . 5
32abbidv 2550 . . . 4
4 dfsbcq2 3164 . . . . 5
54abbidv 2550 . . . 4
63, 5eleq12d 2504 . . 3
7 nfs1v 2182 . . . . . 6
87nfab 2576 . . . . 5
9 nfs1v 2182 . . . . . 6
109nfab 2576 . . . . 5
118, 10nfel 2580 . . . 4
12 sbab 2558 . . . . 5
13 sbab 2558 . . . . 5
1412, 13eleq12d 2504 . . . 4
1511, 14sbie 2149 . . 3
161, 6, 15vtoclbg 3012 . 2
17 df-csb 3252 . . 3
18 df-csb 3252 . . 3
1917, 18eleq12i 2501 . 2
2016, 19syl6bbr 255 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652  wsb 1658   wcel 1725  cab 2422  wsbc 3161  csb 3251 This theorem is referenced by:  sbcnel12g  3268  sbcel1g  3270  sbcel2g  3272  sbccsb2g  3280  fmptdF  24069  csbxpgVD  29006  csbrngVD  29008 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
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