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Theorem rabexd 5334
Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5335. (Contributed by AV, 16-Jul-2019.)
Hypotheses
Ref Expression
rabexd.1 𝐵 = {𝑥𝐴𝜓}
rabexd.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
rabexd (𝜑𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem rabexd
StepHypRef Expression
1 rabexd.1 . 2 𝐵 = {𝑥𝐴𝜓}
2 rabexd.2 . . 3 (𝜑𝐴𝑉)
3 rabexg 5332 . . 3 (𝐴𝑉 → {𝑥𝐴𝜓} ∈ V)
42, 3syl 17 . 2 (𝜑 → {𝑥𝐴𝜓} ∈ V)
51, 4eqeltrid 2838 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475
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-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  rabex2  5335  zorn2lem1  10491  sylow2a  19487  evlslem6  21644  mhpaddcl  21694  mhpvscacl  21697  mretopd  22596  cusgrexilem1  28696  vtxdgf  28728  mntoval  32152  tocycval  32267  prmidlval  32555  evlsvvval  41135  evlsbagval  41138  mhpind  41166  stoweidlem35  44751  stoweidlem50  44766  stoweidlem57  44773  stoweidlem59  44775  subsaliuncllem  45073  subsaliuncl  45074  smflimlem1  45487  smflimlem2  45488  smflimlem3  45489  smflimlem6  45492  smfrec  45505  smfpimcclem  45523  smfsuplem1  45527  smfinflem  45533  smflimsuplem1  45536  smflimsuplem2  45537  smflimsuplem3  45538  smflimsuplem4  45539  smflimsuplem5  45540  smflimsuplem7  45542  fvmptrab  46000  prproropen  46176
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