Theorem csbex 5312

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 5311	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
2		csbex.1	. 2 ⊢ 𝐵 ∈ V
3	1, 2	mpg 1800	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3475  ⦋csb 3894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4324

This theorem is referenced by:  iunopeqop  5522  dfmpo  8088  cantnfdm  9659  cantnff  9669  bpolylem  15992  ruclem1  16174  pcmpt  16825  cidffn  17622  issubc  17785  natffn  17900  fnxpc  18128  evlfcl  18175  odf  19405  selvval  21681  itgfsum  25344  itgparts  25564  vmaf  26623  mulsval  27565  precsexlem3  27655  ttgval  28126  ttgvalOLD  28127  abfmpel  31880  msrf  34533  rdgssun  36259  finxpreclem2  36271  poimirlem17  36505  poimirlem23  36511  poimirlem24  36512  unirep  36582  cdlemk40  39788  aomclem6  41801  rnghmfn  46688  rngchomrnghmresALTV  46894