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Theorem sbcel1v 3624
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel1v ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1v
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3574 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 3340 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 3567 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2815 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb3 2855 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 3395 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 367 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 2034  wcel 2127  Vcvv 3328  [wsbc 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-v 3330  df-sbc 3565
This theorem is referenced by:  tfinds2  7216  filuni  21861  gropeld  26095  grstructeld  26096  f1od2  29779  esum2dlem  30434  bnj110  31206  f1omptsnlem  33465  relowlpssretop  33494  rdgeqoa  33500  cotrclrcl  38505  frege70  38698  frege72  38700  frege91  38719  sbcoreleleq  39216  onfrALTlem4  39229
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