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Theorem sbcel21v 3042
Description: Class substitution into a membership relation. One direction of sbcel2gv 3041 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 2986 . 2 ([𝐵 / 𝑥]𝐴𝑥𝐵 ∈ V)
2 sbcel2gv 3041 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
32biimpd 144 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
41, 3mpcom 36 1 ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  Vcvv 2752  [wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by: (None)
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