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Theorem sbcel2 3831
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel2
Distinct variable group:   ,

Proof of Theorem sbcel2
StepHypRef Expression
1 sbcel12 3823 . . 3
2 csbconstg 3447 . . . 4
32eleq1d 2526 . . 3
41, 3syl5bb 257 . 2
5 sbcex 3337 . . . 4
65con3i 135 . . 3
7 noel 3788 . . . 4
8 csbprc 3821 . . . . 5
98eleq2d 2527 . . . 4
107, 9mtbiri 303 . . 3
116, 102falsed 351 . 2
124, 11pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  e.wcel 1818   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784
This theorem is referenced by:  csbcom  3837  sbccsb  3849  sbnfc2  3854  csbab  3855  sbcssg  3940  csbuni  4277  csbxp  5086  csbdm  5202  issubc  15204  nbgraopALT  24424  csbcom2fi  30534  bj-sbeq  34468  bj-sbceqgALT  34469  bj-sels  34520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785
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