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Theorem sbcel2gv 2886
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel2gv (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcel2gv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2146 . 2 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2 eleq2 2146 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
31, 2sbcie2g 2856 1 (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1434  [wsbc 2824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  sbcel21v  2887  csbvarg  2942
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