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Theorem ssabral 3822
Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
Assertion
Ref Expression
ssabral (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssabral
StepHypRef Expression
1 ssab 3821 . 2 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3066 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2bitr4i 267 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wcel 2145  {cab 2757  wral 3061  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-in 3730  df-ss 3737
This theorem is referenced by:  comppfsc  21556  txdis1cn  21659  qustgplem  22144  xrhmeo  22965  cncmet  23338  itg1addlem4  23686  subfacp1lem6  31505  poimirlem9  33751  istotbnd3  33902  sstotbnd  33906  heibor1lem  33940  heibor1  33941  setis  42972
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