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Theorem sbcel21v 3815
 Description: Class substitution into a membership relation. One direction of sbcel2gv 3814 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel21v ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem sbcel21v
StepHypRef Expression
1 sbcex 3757 . 2 ([𝐵 / 𝑥]𝐴𝑥𝐵 ∈ V)
2 sbcel2gv 3814 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
32biimpd 232 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
41, 3mpcom 38 1 ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3469  [wsbc 3747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sbc 3748 This theorem is referenced by: (None)
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