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Theorem ss2rabi 3417
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 3411 . 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 2768 1  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   {crab 2701    C_ wss 3312
This theorem is referenced by:  supub  7454  suplub  7455  card2on  7512  rankval4  7783  fin1a2lem12  8281  catlid  13898  catrid  13899  gsumval2  14773  lbsextlem3  16222  psrbagsn  16545  musum  20966  ppiub  20978  usisuslgra  21377  lclkrs2  32239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-in 3319  df-ss 3326
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