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

Theorem ssabral 3652
 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 3651 . 2 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 df-ral 2912 . 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 1478   ∈ wcel 1987  {cab 2607  ∀wral 2907   ⊆ wss 3555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-in 3562  df-ss 3569 This theorem is referenced by:  comppfsc  21245  txdis1cn  21348  qustgplem  21834  xrhmeo  22653  cncmet  23027  itg1addlem4  23372  subfacp1lem6  30875  poimirlem9  33050  istotbnd3  33202  sstotbnd  33206  heibor1lem  33240  heibor1  33241
 Copyright terms: Public domain W3C validator