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Theorem ssrabdv 4007
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 3103 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 4006 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 583 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3064  {crab 3068  wss 3887
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904
This theorem is referenced by:  mndind  18466  symggen  19078  ablfac1eu  19676  lspsolvlem  20404  prdsxmslem2  23685  ovolicc2lem4  24684  abelth2  25601  perfectlem2  26378  umgrres1lem  27677  upgrres1  27680  nsgmgc  31597  nsgqusf1olem2  31599  nsgqusf1olem3  31600  cvmlift2lem11  33275  bj-rabtrAUTO  35120  mapdrvallem3  39660  idomsubgmo  41023  k0004ss2  41762  liminfvalxr  43324  smflimlem4  44309  perfectALTVlem2  45174
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