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Theorem ssabral 4059
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 4058 . 2 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3061 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2bitr4i 278 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538  wcel 2105  {cab 2708  wral 3060  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-v 3475  df-in 3955  df-ss 3965
This theorem is referenced by:  comppfsc  23356  txdis1cn  23459  qustgplem  23945  xrhmeo  24791  cncmet  25170  itg1addlem4  25548  itg1addlem4OLD  25549  subfacp1lem6  34641  poimirlem9  36963  istotbnd3  37105  sstotbnd  37109  heibor1lem  37143  heibor1  37144  setis  47907
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