Theorem csbex 5179

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 5178	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
2		csbex.1	. 2 ⊢ 𝐵 ∈ V
3	1, 2	mpg 1799	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:   ∈ wcel 2111  Vcvv 3441  ⦋csb 3828

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-nul 4244

This theorem is referenced by:  iunopeqop  5376  dfmpo  7780  cantnfdm  9111  cantnff  9121  bpolylem  15394  ruclem1  15576  pcmpt  16218  cidffn  16941  issubc  17097  natffn  17211  fnxpc  17418  evlfcl  17464  odf  18657  selvval  20790  itgfsum  24430  itgparts  24650  vmaf  25704  ttgval  26669  abfmpel  30418  msrf  32902  rdgssun  34795  finxpreclem2  34807  poimirlem17  35074  poimirlem23  35080  poimirlem24  35081  unirep  35151  cdlemk40  38213  aomclem6  40003  rnghmfn  44514  rngchomrnghmresALTV  44620