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Theorem sbcel1vOLD 3837
Description: Obsolete version of sbcel1v 3836 as of 30-Apr-2023. (Contributed by NM, 17-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcel1vOLD ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1vOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 3510 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 3772 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2897 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb3 2937 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 3566 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 380 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2060  wcel 2105  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by: (None)
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