Theorem sbcex 3051

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 3043	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elex 2825	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3	1, 2	sylbi 121	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2203  {cab 2218  Vcvv 2813  [wsbc 3042

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-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815  df-sbc 3043

This theorem is referenced by:  sbcco  3064  sbc5  3066  sbcan  3085  sbcor  3087  sbcn1  3090  sbcim1  3091  sbcbi1  3092  sbcal  3094  sbcex2  3096  sbcel1v  3105  sbcel21v  3107  sbcimdv  3108  sbcrext  3120  spesbc  3129  csbprc  3554  opelopabsb  4378