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Theorem ssrab 3941
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 3097 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq2i 3888 . 2 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
3 ssab 3933 . 2 (𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
4 dfss3 3849 . . . 4 (𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
54anbi1i 614 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
6 r19.26 3120 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
7 df-ral 3093 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
85, 6, 73bitr2ri 292 . 2 (∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
92, 3, 83bitri 289 1 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505  wcel 2050  {cab 2758  wral 3088  {crab 3092  wss 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rab 3097  df-in 3838  df-ss 3845
This theorem is referenced by:  ssrabdv  3942  omssnlim  7412  ordtypelem2  8780  ordtypelem10  8788  card2inf  8816  r0weon  9234  ramtlecl  16195  sscntz  18230  ppttop  21322  epttop  21324  cmpcov2  21705  tgcmp  21716  xkoinjcn  22002  fbssfi  22152  filssufilg  22226  uffixfr  22238  tmdgsum2  22411  symgtgp  22416  ghmcnp  22429  blcls  22822  clsocv  23559  lhop1lem  24316  ressatans  25216  axcontlem3  26458  axcontlem4  26459  ldgenpisyslem3  31069  ldgenpisys  31070  imambfm  31165  connpconn  32067  cvmlift2lem11  32145  cvmlift2lem12  32146  bj-rabtr  33743  hbtlem6  39125
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