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 3475	. 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 2147  {cab 2741  Vcvv 3454  [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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  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  sbcfal  3816  sbccomlem  3821  sbcrext  3824  sbcreu  3827  spesbc  3833  csbprc  4370  sbcel12  4372  sbcne12  4376  sbcel2  4379  sbccsb2  4398  sbcbr123  5164  opelopabsb  5509  csbopab  5535  csbxp  5758  csbiota  6554  csbriota  7418  fi1uzind  14682  bj-csbprc  38013  sbccomieg  44004