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Theorem sbcel2gv 2924
 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 2163 . 2 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2 eleq2 2163 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
31, 2sbcie2g 2894 1 (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 1448  [wsbc 2862 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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863 This theorem is referenced by:  sbcel21v  2925  csbvarg  2980
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