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Theorem ssrabdv 3644
 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 2949 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 3643 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 695 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-in 3547  df-ss 3554 This theorem is referenced by:  mrcmndind  17189  symggen  17713  ablfac1eu  18295  lspsolvlem  18963  prdsxmslem2  22144  ovolicc2lem4  23095  abelth2  24000  perfectlem2  24755  cvmlift2lem11  30549  bj-rabtrAUTO  32121  mapdrvallem3  35953  idomsubgmo  36795  k0004ss2  37470  smflimlem4  39660  perfectALTVlem2  40165  umgrres1lem  40529  upgrres1  40532
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