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Theorem sbcel2gv 3602
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 2792 . 2 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2 eleq2 2792 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
31, 2sbcie2g 3575 1 (𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2103  [wsbc 3541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-v 3306  df-sbc 3542
This theorem is referenced by:  sbcel21v  3603  csbvarg  4111  bnj92  31160  bnj539  31189  frege77  38653  sbcoreleleq  39164  trsbc  39169  onfrALTlem5  39176  sbcoreleleqVD  39511  trsbcVD  39529  onfrALTlem5VD  39537
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