MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ss2rabi Unicode version

Theorem ss2rabi 3370
Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
ss2rabi.1  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
ss2rabi  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }

Proof of Theorem ss2rabi
StepHypRef Expression
1 ss2rab 3364 . 2  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<-> 
A. x  e.  A  ( ph  ->  ps )
)
2 ss2rabi.1 . 2  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
31, 2mprgbir 2721 1  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   {crab 2655    C_ wss 3265
This theorem is referenced by:  supub  7399  suplub  7400  card2on  7457  rankval4  7728  fin1a2lem12  8226  catlid  13837  catrid  13838  gsumval2  14712  lbsextlem3  16161  psrbagsn  16484  musum  20845  ppiub  20857  usisuslgra  21256  itgaddnclem2  25966  lclkrs2  31657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rab 2660  df-in 3272  df-ss 3279
  Copyright terms: Public domain W3C validator