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Theorem sbcex 2846
Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcex ([𝐴 / 𝑥]𝜑𝐴 ∈ V)

Proof of Theorem sbcex
StepHypRef Expression
1 df-sbc 2839 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
2 elex 2630 . 2 (𝐴 ∈ {𝑥𝜑} → 𝐴 ∈ V)
31, 2sylbi 119 1 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  {cab 2074  Vcvv 2619  [wsbc 2838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2839
This theorem is referenced by:  sbcco  2859  sbc5  2861  sbcan  2879  sbcor  2881  sbcn1  2884  sbcim1  2885  sbcbi1  2886  sbcal  2888  sbcex2  2890  sbcel1v  2899  sbcel21v  2901  sbcimdv  2902  sbcrext  2914  spesbc  2922  csbprc  3325  opelopabsb  4078
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