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Theorem sbcel2gv 3863
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 2828 . 2 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2 eleq2 2828 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
31, 2sbcie2g 3834 1 (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  sbcel21v  3864  csbvarg  4440  bnj92  34855  bnj539  34884  minregex  43524  frege77  43930  sbcoreleleq  44533  trsbc  44538  onfrALTlem5  44540  sbcoreleleqVD  44857  trsbcVD  44875  onfrALTlem5VD  44883  modelaxreplem3  44945
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