Theorem ssrabdv 4019

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 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
4		ssrab 4018	. 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 395   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396  df-ss 3914

This theorem is referenced by:  mndind  18731  symggen  19377  ablfac1eu  19982  lspsolvlem  21074  prdsxmslem2  24439  ovolicc2lem4  25443  abelth2  26374  perfectlem2  27163  umgrres1lem  29283  upgrres1  29286  nsgmgc  33369  nsgqusf1olem2  33371  nsgqusf1olem3  33372  cvmlift2lem11  35349  bj-rabtrAUTO  36966  mapdrvallem3  41685  idomsubgmo  43226  nadd2rabtr  43417  k0004ss2  44185  liminfvalxr  45821  smflimlem4  46812  perfectALTVlem2  47753