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Theorem sbcex 3733
 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 3724 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
2 elex 3462 . 2 (𝐴 ∈ {𝑥𝜑} → 𝐴 ∈ V)
31, 2sylbi 220 1 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  {cab 2779  Vcvv 3444  [wsbc 3723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724 This theorem is referenced by:  sbccow  3746  sbcco  3749  sbc5  3751  sbcan  3771  sbcor  3772  sbcn1  3774  sbcim1  3775  sbcbi1  3780  sbcal  3783  sbcex2  3784  sbcel1v  3789  sbcel21v  3791  sbcimdv  3792  sbcrext  3805  sbcreu  3808  spesbc  3814  csbprc  4316  sbcel12  4319  sbcne12  4323  sbcel2  4326  sbccsb2  4345  sbcbr123  5087  opelopabsb  5385  csbopab  5410  csbxp  5618  csbiota  6321  csbriota  7112  fi1uzind  13855  bj-csbprc  34351  sbccomieg  39727
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