Theorem ssrabdv 4071

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
ssrabdv.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)

Assertion

Ref	Expression
ssrabdv	⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
2		ssrabdv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)
3	2	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
4		ssrab 4070	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 3, 4	sylanbrc 583	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3061  {crab 3432   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rab 3433  df-v 3476  df-in 3955  df-ss 3965

This theorem is referenced by:  mndind  18708  symggen  19337  ablfac1eu  19942  lspsolvlem  20754  prdsxmslem2  24037  ovolicc2lem4  25036  abelth2  25953  perfectlem2  26730  umgrres1lem  28564  upgrres1  28567  nsgmgc  32518  nsgqusf1olem2  32520  nsgqusf1olem3  32521  cvmlift2lem11  34299  bj-rabtrAUTO  35807  mapdrvallem3  40512  idomsubgmo  41930  nadd2rabtr  42124  k0004ss2  42893  liminfvalxr  44489  smflimlem4  45480  perfectALTVlem2  46380