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Theorem sbcex 2772
 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 2765 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
2 elex 2566 . 2 (𝐴 ∈ {𝑥𝜑} → 𝐴 ∈ V)
31, 2sylbi 114 1 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  {cab 2026  Vcvv 2557  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-sbc 2765 This theorem is referenced by:  sbcco  2785  sbc5  2787  sbcan  2805  sbcor  2807  sbcal  2810  sbcex2  2812  sbcrext  2835  spesbc  2843  csbprc  3262  opelopabsb  3997
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