Theorem rabssdv 4098

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 1119	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵)))
3	2	ralrimiv 3151	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
4		rabss 4095	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
5	3, 4	sylibr 234	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-ss 3993

This theorem is referenced by:  suppss2  8241  oemapvali  9753  cantnflem1  9758  harval2  10066  zsupss  13002  ramub1lem1  17073  symggen  19512  efgsfo  19781  ablfacrp  20110  ablfac1eu  20117  pgpfac1lem5  20123  ablfaclem3  20131  nrmr0reg  23778  ptcmplem3  24083  abelthlem2  26494  lgamgulmlem1  27090  sltonold  28301  rspectopn  33813  neibastop2lem  36326  topmeet  36330  weiunse  36434  cntotbnd  37756  mapdrvallem2  41602  aks6d1c6lem3  42129  onintunirab  43188  nadd2rabex  43348  k0004ss1  44113  liminfvalxr  45704