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Theorem inex1g 4761
Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
inex1g (𝐴𝑉 → (𝐴𝐵) ∈ V)

Proof of Theorem inex1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3785 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2683 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 vex 3189 . . 3 𝑥 ∈ V
43inex1 4759 . 2 (𝑥𝐵) ∈ V
52, 4vtoclg 3252 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562
This theorem is referenced by:  dmresexg  5380  onin  5713  offval  6857  offval3  7107  onsdominel  8053  ssenen  8078  inelfi  8268  fiin  8272  tskwe  8720  dfac8b  8798  ac10ct  8801  infpwfien  8829  fictb  9011  canthnum  9415  gruina  9584  ressinbas  15857  ressress  15859  qusin  16125  catcbas  16668  fpwipodrs  17085  psss  17135  gsumzres  18231  eltg  20672  eltg3  20677  ntrval  20750  restco  20878  restfpw  20893  ordtrest  20916  ordtrest2lem  20917  ordtrest2  20918  cnrmi  21074  restcnrm  21076  kgeni  21250  tsmsfbas  21841  eltsms  21846  tsmsres  21857  caussi  23003  causs  23004  elpwincl1  29201  disjdifprg2  29231  sigainb  29977  ldgenpisyslem1  30004  carsgclctun  30161  eulerpartlemgs2  30220  sseqval  30228  bnj1177  30779  cvmsss2  30961  fnemeet2  32001  ontgval  32069  bj-diagval  32720  fin2so  33025  elrfi  36734  iunrelexp0  37472  fourierdlem71  39698  fourierdlem80  39707  sge0less  39913  sge0ssre  39918  carageniuncllem2  40040  dfrngc2  41257  rnghmsscmap2  41258  rngcbasALTV  41268  dfringc2  41303  rhmsscmap2  41304  rhmsscrnghm  41311  rngcresringcat  41315  ringcbasALTV  41331  srhmsubc  41361  fldc  41368  fldhmsubc  41369  rngcrescrhm  41370  srhmsubcALTV  41379  fldcALTV  41386  fldhmsubcALTV  41387  rngcrescrhmALTV  41388  offval0  41584
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