Theorem sbcex 2921

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]𝜑 → 𝐴 ∈ V)

Proof of Theorem sbcex

Step	Hyp	Ref	Expression
1		df-sbc 2914	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elex 2700	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3	1, 2	sylbi 120	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1481  {cab 2126  Vcvv 2689  [wsbc 2913

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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-sbc 2914

This theorem is referenced by:  sbcco  2934  sbc5  2936  sbcan  2955  sbcor  2957  sbcn1  2960  sbcim1  2961  sbcbi1  2962  sbcal  2964  sbcex2  2966  sbcel1v  2975  sbcel21v  2977  sbcimdv  2978  sbcrext  2990  spesbc  2998  csbprc  3413  opelopabsb  4190