Theorem csbex 5251

Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by NM, 17-Aug-2018.)

Hypothesis

Ref	Expression
csbex.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
csbex	⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Proof of Theorem csbex

Step	Hyp	Ref	Expression
1		csbexg 5250	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
2		csbex.1	. 2 ⊢ 𝐵 ∈ V
3	1, 2	mpg 1807	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:   ∈ wcel 2132  Vcvv 3444  ⦋csb 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-nul 5246

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-nul 4277

This theorem is referenced by:  iunopeqop  5480  iunopeqopOLD  5481  dfmpo  8065  cantnfdm  9605  cantnff  9615  bpolylem  16050  ruclem1  16235  pcmpt  16900  cidffn  17682  issubc  17840  natffn  17957  fnxpc  18180  evlfcl  18226  odf  19549  rnghmfn  20456  selvval  22142  itgfsum  25858  itgparts  26078  vmaf  27149  mulsval  28168  precsexlem3  28268  ttgval  29010  abfmpel  32796  msrf  35830  rdgssun  37810  finxpreclem2  37822  poimirlem17  38074  poimirlem23  38080  poimirlem24  38081  unirep  38151  cdlemk40  41479  aomclem6  43574  rngchomrnghmresALTV  48839  idfurcl  49657  fucofn2  49883  dfinito4  50060  dftermo4  50061  lanfn  50168  ranfn  50169