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Theorem ss2rab 3044
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 2332 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2332 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2sseq12i 2999 . 2  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<->  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  ( x  e.  A  /\  ps ) } )
4 ss2ab 3036 . 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 2328 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
6 imdistan 426 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
76albii 1375 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps ) ) )
85, 7bitr2i 178 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  <->  A. x  e.  A  ( ph  ->  ps )
)
93, 4, 83bitri 199 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 101    <-> wb 102   A.wal 1257    e. wcel 1409   {cab 2042   A.wral 2323   {crab 2327    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-in 2952  df-ss 2959
This theorem is referenced by:  ss2rabdv  3049  ss2rabi  3050
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