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Theorem sbcel12 3823
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel12

Proof of Theorem sbcel12
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3330 . . . 4
2 dfsbcq2 3330 . . . . . 6
32abbidv 2593 . . . . 5
4 dfsbcq2 3330 . . . . . 6
54abbidv 2593 . . . . 5
63, 5eleq12d 2539 . . . 4
7 nfs1v 2181 . . . . . . 7
87nfab 2623 . . . . . 6
9 nfs1v 2181 . . . . . . 7
109nfab 2623 . . . . . 6
118, 10nfel 2632 . . . . 5
12 sbab 2604 . . . . . 6
13 sbab 2604 . . . . . 6
1412, 13eleq12d 2539 . . . . 5
1511, 14sbie 2149 . . . 4
161, 6, 15vtoclbg 3168 . . 3
17 df-csb 3435 . . . 4
18 df-csb 3435 . . . 4
1917, 18eleq12i 2536 . . 3
2016, 19syl6bbr 263 . 2
21 sbcex 3337 . . . 4
2221con3i 135 . . 3
23 noel 3788 . . . 4
24 csbprc 3821 . . . . 5
2524eleq2d 2527 . . . 4
2623, 25mtbiri 303 . . 3
2722, 262falsed 351 . 2
2820, 27pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  {cab 2442   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784
This theorem is referenced by:  sbcnel12g  3826  sbcel1g  3829  sbcel2  3831  sbccsb2  3851  csbmpt12  4786  ixpsnval  7492  fmptdF  27495  finixpnum  30038
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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