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Theorem rabss 3142
Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
rabss  |-  ( { x  e.  A  |  ph }  C_  B  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rabss
StepHypRef Expression
1 df-rab 2400 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq1i 3091 . 2  |-  ( { x  e.  A  |  ph }  C_  B  <->  { x  |  ( x  e.  A  /\  ph ) }  C_  B )
3 abss 3134 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  C_  B  <->  A. x
( ( x  e.  A  /\  ph )  ->  x  e.  B ) )
4 impexp 261 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  ->  x  e.  B )  <->  ( x  e.  A  ->  ( ph  ->  x  e.  B ) ) )
54albii 1429 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  e.  B
)  <->  A. x ( x  e.  A  ->  ( ph  ->  x  e.  B
) ) )
6 df-ral 2396 . . 3  |-  ( A. x  e.  A  ( ph  ->  x  e.  B
)  <->  A. x ( x  e.  A  ->  ( ph  ->  x  e.  B
) ) )
75, 6bitr4i 186 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  e.  B
)  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
82, 3, 73bitri 205 1  |-  ( { x  e.  A  |  ph }  C_  B  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    e. wcel 1463   {cab 2101   A.wral 2391   {crab 2395    C_ wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-in 3045  df-ss 3052
This theorem is referenced by:  rabssdv  3145  dvdsssfz1  11457  phibndlem  11798  dfphi2  11802  istopon  12086  blsscls2  12568
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