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Theorem sbcel1v 3861
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2374. (Revised by Wolf Lammen, 30-Apr-2023.)
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 3800 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 3498 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 3793 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2826 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb1 2865 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 3556 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 378 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2061  wcel 2105  Vcvv 3477  [wsbc 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-sbc 3791
This theorem is referenced by:  tfinds2  7884  filuni  23908  gropeld  29064  grstructeld  29065  f1od2  32738  esum2dlem  34072  bnj110  34850  f1omptsnlem  37318  relowlpssretop  37346  rdgeqoa  37352  minregex  43523  cotrclrcl  43731  frege70  43922  frege72  43924  frege91  43943  sbcoreleleq  44532  onfrALTlem4  44540  sbcoreleleqVD  44856  onfrALTlem4VD  44883  rspesbcd  44935
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