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Theorem sbcel2gv 3463
 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 2677 . 2 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2 eleq2 2677 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
31, 2sbcie2g 3436 1 (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  sbcel21v  3464  csbvarg  3955  bnj92  30186  bnj539  30215  frege77  37254  sbcoreleleq  37766  trsbc  37771  onfrALTlem5  37778  sbcoreleleqVD  38117  trsbcVD  38135  onfrALTlem5VD  38143
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