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Theorem ssrab 4066
Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 3420 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq2i 4008 . 2 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
3 ssab 4057 . 2 (𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
4 dfss3 3967 . . . 4 (𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
54anbi1i 622 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
6 r19.26 3101 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
7 df-ral 3052 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
85, 6, 73bitr2ri 299 . 2 (∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
92, 3, 83bitri 296 1 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532  wcel 2099  {cab 2703  wral 3051  {crab 3419  wss 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rab 3420  df-ss 3963
This theorem is referenced by:  ssrabdv  4067  omssnlim  7883  ordtypelem2  9555  ordtypelem10  9563  card2inf  9591  r0weon  10048  ramtlecl  16997  sscntz  19316  ppttop  22998  epttop  23000  cmpcov2  23382  tgcmp  23393  xkoinjcn  23679  fbssfi  23829  filssufilg  23903  uffixfr  23915  tmdgsum2  24088  symgtgp  24098  ghmcnp  24107  blcls  24503  clsocv  25266  lhop1lem  26034  ressatans  26959  axcontlem3  28897  axcontlem4  28898  ldgenpisyslem3  34011  ldgenpisys  34012  imambfm  34109  lfuhgr  34958  connpconn  35076  cvmlift2lem11  35154  cvmlift2lem12  35155  bj-rabtr  36649  hbtlem6  42827
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