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Theorem sbcel21v 2917
Description: Class substitution into a membership relation. One direction of sbcel2gv 2916 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 2862 . 2 ([𝐵 / 𝑥]𝐴𝑥𝐵 ∈ V)
2 sbcel2gv 2916 . . 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 1445  Vcvv 2633  [wsbc 2854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-sbc 2855
This theorem is referenced by: (None)
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