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Theorem ssabral 3870
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 3869 . 2 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3095 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2bitr4i 270 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wcel 2157  {cab 2786  wral 3090  wss 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-in 3777  df-ss 3784
This theorem is referenced by:  comppfsc  21663  txdis1cn  21766  qustgplem  22251  xrhmeo  23072  cncmet  23447  itg1addlem4  23806  subfacp1lem6  31683  poimirlem9  33906  istotbnd3  34056  sstotbnd  34060  heibor1lem  34094  heibor1  34095  setis  43238
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