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Theorem sbcel21v 3001
Description: Class substitution into a membership relation. One direction of sbcel2gv 3000 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 2945 . 2 ([𝐵 / 𝑥]𝐴𝑥𝐵 ∈ V)
2 sbcel2gv 3000 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
32biimpd 143 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
41, 3mpcom 36 1 ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  Vcvv 2712  [wsbc 2937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938
This theorem is referenced by: (None)
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