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Theorem sbcel1gvOLD 39616
 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 3646 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 3590 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
2 eleq1 2838 . 2 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
3 clelsb3 2878 . 2 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
41, 2, 3vtoclbg 3418 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  [wsb 2049   ∈ wcel 2145  [wsbc 3587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588 This theorem is referenced by:  sbcoreleleqVD  39617  onfrALTlem4VD  39644
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