Theorem csbex 5317

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 5316	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
2		csbex.1	. 2 ⊢ 𝐵 ∈ V
3	1, 2	mpg 1794	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478  ⦋csb 3908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340

This theorem is referenced by:  iunopeqop  5531  dfmpo  8126  cantnfdm  9702  cantnff  9712  bpolylem  16081  ruclem1  16264  pcmpt  16926  cidffn  17723  issubc  17886  natffn  18004  fnxpc  18232  evlfcl  18279  odf  19570  rnghmfn  20456  selvval  22157  itgfsum  25877  itgparts  26103  vmaf  27177  mulsval  28150  precsexlem3  28248  ttgval  28898  ttgvalOLD  28899  abfmpel  32672  msrf  35527  rdgssun  37361  finxpreclem2  37373  poimirlem17  37624  poimirlem23  37630  poimirlem24  37631  unirep  37701  cdlemk40  40900  aomclem6  43048  rngchomrnghmresALTV  48123