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Theorem ssrabdv 4074
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1 (𝜑𝐵𝐴)
ssrabdv.2 ((𝜑𝑥𝐵) → 𝜓)
Assertion
Ref Expression
ssrabdv (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2 (𝜑𝐵𝐴)
2 ssrabdv.2 . . 3 ((𝜑𝑥𝐵) → 𝜓)
32ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 4073 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 583 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  {crab 3436  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3968
This theorem is referenced by:  mndind  18841  symggen  19488  ablfac1eu  20093  lspsolvlem  21144  prdsxmslem2  24542  ovolicc2lem4  25555  abelth2  26486  perfectlem2  27274  umgrres1lem  29327  upgrres1  29330  nsgmgc  33440  nsgqusf1olem2  33442  nsgqusf1olem3  33443  cvmlift2lem11  35318  bj-rabtrAUTO  36933  mapdrvallem3  41648  idomsubgmo  43205  nadd2rabtr  43397  k0004ss2  44165  liminfvalxr  45798  smflimlem4  46789  perfectALTVlem2  47709
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