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Theorem sbcel1gvOLD 38116
 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 3462 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 3405 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
2 eleq1 2676 . 2 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
3 clelsb3 2716 . 2 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
41, 2, 3vtoclbg 3240 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  [wsb 1867   ∈ 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-11 2021  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:  sbcoreleleqVD  38117  onfrALTlem4VD  38144
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