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Theorem ssrabdv 3872
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 3150 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 3871 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 574 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  wral 3092  {crab 3096  wss 3763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rab 3101  df-in 3770  df-ss 3777
This theorem is referenced by:  mrcmndind  17565  symggen  18085  ablfac1eu  18668  lspsolvlem  19344  prdsxmslem2  22541  ovolicc2lem4  23495  abelth2  24404  perfectlem2  25163  umgrres1lem  26412  upgrres1  26415  cvmlift2lem11  31612  bj-rabtrAUTO  33234  mapdrvallem3  37421  idomsubgmo  38271  k0004ss2  38944  liminfvalxr  40489  smflimlem4  41458  perfectALTVlem2  42200
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