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Theorem ssrabdv 4032
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 3140 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 4031 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 584 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  {crab 3406  wss 3911
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  mndind  18643  symggen  19257  ablfac1eu  19857  lspsolvlem  20619  prdsxmslem2  23901  ovolicc2lem4  24900  abelth2  25817  perfectlem2  26594  umgrres1lem  28300  upgrres1  28303  nsgmgc  32238  nsgqusf1olem2  32240  nsgqusf1olem3  32241  cvmlift2lem11  33964  bj-rabtrAUTO  35448  mapdrvallem3  40155  idomsubgmo  41568  nadd2rabtr  41743  k0004ss2  42512  liminfvalxr  44110  smflimlem4  45101  perfectALTVlem2  46000
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