Theorem ssrabdv 4027

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 3121	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
4		ssrab 4026	. 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 2109  ∀wral 3044  {crab 3396   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3397  df-ss 3922

This theorem is referenced by:  mndind  18721  symggen  19368  ablfac1eu  19973  lspsolvlem  21068  prdsxmslem2  24434  ovolicc2lem4  25438  abelth2  26369  perfectlem2  27158  umgrres1lem  29274  upgrres1  29277  nsgmgc  33368  nsgqusf1olem2  33370  nsgqusf1olem3  33371  cvmlift2lem11  35305  bj-rabtrAUTO  36925  mapdrvallem3  41645  idomsubgmo  43186  nadd2rabtr  43377  k0004ss2  44145  liminfvalxr  45784  smflimlem4  46775  perfectALTVlem2  47726