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Theorem ss2rab 3086
Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
ss2rab  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<-> 
A. x  e.  A  ( ph  ->  ps )
)

Proof of Theorem ss2rab
StepHypRef Expression
1 df-rab 2364 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2364 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2sseq12i 3041 . 2  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<->  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  A  /\  ps ) } )
4 ss2ab 3078 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  A  /\  ps ) } 
<-> 
A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
5 df-ral 2360 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
6 imdistan 433 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
76albii 1402 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
85, 7bitr2i 183 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  <->  A. x  e.  A  ( ph  ->  ps )
)
93, 4, 83bitri 204 1  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<-> 
A. x  e.  A  ( ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1285    e. wcel 1436   {cab 2071   A.wral 2355   {crab 2359    C_ wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-in 2994  df-ss 3001
This theorem is referenced by:  ss2rabdv  3091  ss2rabi  3092
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