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Theorem rabss2 3180
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 pm3.45 586 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
21alimi 1431 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
3 dfss2 3086 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 ss2ab 3165 . . 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 200 . 2  |-  ( A 
C_  B  ->  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  B  /\  ph ) } )
6 df-rab 2425 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
7 df-rab 2425 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
85, 6, 73sstr4g 3140 1  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  C_ 
{ x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1329    e. wcel 1480   {cab 2125   {crab 2420    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-in 3077  df-ss 3084
This theorem is referenced by:  sess2  4260  dvfgg  12836
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