Theorem sbcex 3739

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 3730	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elex 3451	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3	1, 2	sylbi 217	1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  {cab 2715  Vcvv 3430  [wsbc 3729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-sbc 3730

This theorem is referenced by:  sbccow  3752  sbcco  3755  sbc5ALT  3758  sbcan  3779  sbcor  3780  sbcn1  3782  sbcim1  3783  sbcbi1  3787  sbcal  3789  sbcex2  3790  sbcel1v  3795  sbcel21v  3797  sbccomlem  3808  sbcrext  3812  sbcreu  3815  spesbc  3821  csbprc  4350  sbcel12  4352  sbcne12  4356  sbcel2  4359  sbccsb2  4378  sbcbr123  5140  opelopabsb  5478  csbopab  5503  csbxp  5725  csbiota  6485  csbriota  7332  fi1uzind  14460  bj-csbprc  37233  sbccomieg  43239