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Theorem sbcel21v 3483
Description: Class substitution into a membership relation. One direction of sbcel2gv 3482 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 3431 . 2 ([𝐵 / 𝑥]𝐴𝑥𝐵 ∈ V)
2 sbcel2gv 3482 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
32biimpd 219 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
41, 3mpcom 38 1 ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3189  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191  df-sbc 3422
This theorem is referenced by: (None)
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