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Theorem ssrabdv 3822
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 3104 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
4 ssrab 3821 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
51, 3, 4sylanbrc 701 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  {crab 3054  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-in 3722  df-ss 3729
This theorem is referenced by:  mrcmndind  17587  symggen  18110  ablfac1eu  18692  lspsolvlem  19364  prdsxmslem2  22555  ovolicc2lem4  23508  abelth2  24415  perfectlem2  25175  umgrres1lem  26422  upgrres1  26425  cvmlift2lem11  31623  bj-rabtrAUTO  33254  mapdrvallem3  37455  idomsubgmo  38296  k0004ss2  38970  liminfvalxr  40536  smflimlem4  41506  perfectALTVlem2  42159
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