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Theorem rabss2 3078
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 pm3.45 562 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
21alimi 1385 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
3 dfss2 2989 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 ss2ab 3063 . . 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 199 . 2  |-  ( A 
C_  B  ->  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  B  /\  ph ) } )
6 df-rab 2358 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
7 df-rab 2358 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
85, 6, 73sstr4g 3041 1  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  C_ 
{ x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    e. wcel 1434   {cab 2068   {crab 2353    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-in 2980  df-ss 2987
This theorem is referenced by:  sess2  4101
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