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Theorem sbcel1v 3017
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel1v ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1v
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2963 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 2741 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 2958 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2233 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb1 2275 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 2791 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 699 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1755  wcel 2141  Vcvv 2730  [wsbc 2955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956
This theorem is referenced by:  f1od2  6211
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