Theorem sbcex 3752

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 3743	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elex 3474	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3	1, 2	sylbi 220	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  {cab 2740  Vcvv 3453  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743

This theorem is used by:  sbccow  3765  sbcco  3768  sbc5ALT  3771  sbcan  3791  sbcor  3792  sbcn1  3794  sbcim1  3795  sbcbi1  3799  sbcal  3801  sbcex2  3802  sbcel1v  3807  sbcel21v  3809  sbccomlem  3820  sbcrext  3823  sbcreu  3826  spesbc  3832  csbprc  4370  sbcel12  4372  sbcne12  4376  sbcel2  4379  sbccsb2  4398  sbcbr123  5163  opelopabsb  5512  csbopab  5538  csbxp  5760  csbiota  6530  csbriota  7389  fi1uzind  14576  bj-csbprc  37661  sbccomieg  43642