Theorem rabssdv 4068

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)

Hypothesis

Ref	Expression
rabssdv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵)

Assertion

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

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rabssdv

Step	Hyp	Ref	Expression
1		rabssdv.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵)
2	1	3exp 1116	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵)))
3	2	ralrimiv 3134	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
4		rabss 4065	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
5	3, 4	sylibr 233	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1084   ∈ wcel 2098  ∀wral 3050  {crab 3418   ⊆ wss 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rab 3419  df-ss 3961

This theorem is referenced by:  suppss2  8206  oemapvali  9714  cantnflem1  9719  harval2  10027  zsupss  12959  ramub1lem1  17014  symggen  19454  efgsfo  19723  ablfacrp  20052  ablfac1eu  20059  pgpfac1lem5  20065  ablfaclem3  20073  nrmr0reg  23714  ptcmplem3  24019  abelthlem2  26431  lgamgulmlem1  27026  sltonold  28223  rspectopn  33619  neibastop2lem  35995  topmeet  35999  cntotbnd  37420  mapdrvallem2  41268  aks6d1c6lem3  41794  onintunirab  42802  nadd2rabex  42962  k0004ss1  43728  liminfvalxr  45314