Theorem ssrabdv 3973

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 3097	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
4		ssrab 3972	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 3, 4	sylanbrc 586	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  ∀wral 3054  {crab 3058   ⊆ wss 3853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rab 3063  df-v 3402  df-in 3860  df-ss 3870

This theorem is referenced by:  mndind  18121  symggen  18729  ablfac1eu  19327  lspsolvlem  20046  prdsxmslem2  23295  ovolicc2lem4  24285  abelth2  25202  perfectlem2  25979  umgrres1lem  27265  upgrres1  27268  nsgmgc  31182  nsgqusf1olem2  31184  nsgqusf1olem3  31185  cvmlift2lem11  32859  bj-rabtrAUTO  34776  mapdrvallem3  39316  idomsubgmo  40636  k0004ss2  41349  liminfvalxr  42907  smflimlem4  43889  perfectALTVlem2  44756