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Theorem ssrabdv 4083
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 3143 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 4082 . 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 2105  wral 3058  {crab 3432  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rab 3433  df-ss 3979
This theorem is referenced by:  mndind  18853  symggen  19502  ablfac1eu  20107  lspsolvlem  21161  prdsxmslem2  24557  ovolicc2lem4  25568  abelth2  26500  perfectlem2  27288  umgrres1lem  29341  upgrres1  29344  nsgmgc  33419  nsgqusf1olem2  33421  nsgqusf1olem3  33422  cvmlift2lem11  35297  bj-rabtrAUTO  36914  mapdrvallem3  41628  idomsubgmo  43181  nadd2rabtr  43373  k0004ss2  44141  liminfvalxr  45738  smflimlem4  46729  perfectALTVlem2  47646
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