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Theorem rabss2 3544
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rabss2  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  C_ 
{ x  e.  B  |  ph } )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem rabss2
StepHypRef Expression
1 pm3.45 842 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
21alimi 1678 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
3 dfss2 3453 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 ss2ab 3529 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  B  /\  ph ) } 
<-> 
A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph )
) )
52, 3, 43imtr4i 269 . 2  |-  ( A 
C_  B  ->  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  B  /\  ph ) } )
6 df-rab 2780 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
7 df-rab 2780 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
85, 6, 73sstr4g 3505 1  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  C_ 
{ x  e.  B  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435    e. wcel 1872   {cab 2407   {crab 2775    C_ wss 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rab 2780  df-in 3443  df-ss 3450
This theorem is referenced by:  sess2  4822  suppfnss  6951  hashbcss  14955  dprdss  17661  minveclem4  22372  minveclem4OLD  22384  prmdvdsfi  24032  mumul  24106  sqff1o  24107  rpvmasumlem  24323  disjxwwlkn  25471  clwwlknfi  25504  shatomistici  28012  rabfodom  28139  xpinpreima2  28721  ballotth  29378  ballotthOLD  29416  bj-unrab  31498  icorempt2  31718  lssats  32547  lpssat  32548  lssatle  32550  lssat  32551  atlatmstc  32854  dochspss  34915  rmxyelqirr  35728  idomodle  36040
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