Theorem sbcex 3786

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 3777	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elex 3482	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3	1, 2	sylbi 216	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099  {cab 2703  Vcvv 3462  [wsbc 3776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-sbc 3777

This theorem is referenced by:  sbccow  3799  sbcco  3802  sbc5ALT  3805  sbcan  3829  sbcor  3830  sbcn1  3832  sbcim1  3833  sbcim1OLD  3834  sbcbi1  3838  sbcal  3840  sbcex2  3841  sbcel1v  3847  sbcel21v  3849  sbcimdvOLD  3851  sbcrext  3866  sbcreu  3869  spesbc  3875  csbprc  4411  sbcel12  4413  sbcne12  4417  sbcel2  4420  sbccsb2  4439  sbcbr123  5207  opelopabsb  5536  csbopab  5561  csbxp  5781  csbiota  6547  csbriota  7396  fi1uzind  14516  bj-csbprc  36616  sbccomieg  42450