MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssabral Structured version   Visualization version   GIF version

Theorem ssabral 4074
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 4073 . 2 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3059 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2bitr4i 278 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wcel 2105  {cab 2711  wral 3058  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-ss 3979
This theorem is referenced by:  comppfsc  23555  txdis1cn  23658  qustgplem  24144  xrhmeo  24990  cncmet  25369  itg1addlem4  25747  itg1addlem4OLD  25748  subfacp1lem6  35169  poimirlem9  37615  istotbnd3  37757  sstotbnd  37761  heibor1lem  37795  heibor1  37796  setis  48928
  Copyright terms: Public domain W3C validator