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Theorem rabexd 4774
Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4775. (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 4772 . . 3 (𝐴𝑉 → {𝑥𝐴𝜓} ∈ V)
42, 3syl 17 . 2 (𝜑 → {𝑥𝐴𝜓} ∈ V)
51, 4syl5eqel 2702 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186
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-rab 2916  df-v 3188  df-in 3562  df-ss 3569
This theorem is referenced by:  rabex2  4775  rabex2OLD  4777  zorn2lem1  9262  sylow2a  17955  evlslem6  19432  mretopd  20806  cusgrexilem1  26222  vtxdgf  26253  stoweidlem35  39559  stoweidlem50  39574  stoweidlem57  39581  stoweidlem59  39583  subsaliuncllem  39882  subsaliuncl  39883  smflimlem1  40286  smflimlem2  40287  smflimlem3  40288  smflimlem6  40291  smfrec  40303  smfpimcclem  40320  smfsuplem1  40324  smfinflem  40330  smflimsuplem1  40333  smflimsuplem2  40334  smflimsuplem3  40335  smflimsuplem4  40336  smflimsuplem5  40337  smflimsuplem7  40339
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