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Theorem ssrabdv 3830
Description: Subclass of a restricted class abstraction (deduction rule). (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 3115 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 3829 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 572 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wral 3061  {crab 3065  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-in 3730  df-ss 3737
This theorem is referenced by:  mrcmndind  17574  symggen  18097  ablfac1eu  18680  lspsolvlem  19356  prdsxmslem2  22554  ovolicc2lem4  23508  abelth2  24416  perfectlem2  25176  umgrres1lem  26425  upgrres1  26428  clwwlknonclwlknonf1olemOLD  27550  cvmlift2lem11  31633  bj-rabtrAUTO  33261  mapdrvallem3  37456  idomsubgmo  38302  k0004ss2  38976  liminfvalxr  40533  smflimlem4  41502  perfectALTVlem2  42159
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