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Theorem sbcel1gvOLD 38914
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 17-Aug-2018. Use sbcel1v 3489 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcel1gvOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem sbcel1gvOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3432 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
2 eleq1 2687 . 2 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
3 clelsb3 2727 . 2 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
41, 2, 3vtoclbg 3262 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1878  wcel 1988  [wsbc 3429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430
This theorem is referenced by:  sbcoreleleqVD  38915  onfrALTlem4VD  38942
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