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Theorem sbcel1v 3025
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 2971 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 2748 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 2965 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2240 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb1 2282 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 2798 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 704 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  [wsb 1762  wcel 2148  Vcvv 2737  [wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by:  f1od2  6235
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