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Theorem ss2ab 3073
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
Assertion
Ref Expression
ss2ab  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( ph  ->  ps ) )

Proof of Theorem ss2ab
StepHypRef Expression
1 nfab1 2225 . . 3  |-  F/_ x { x  |  ph }
2 nfab1 2225 . . 3  |-  F/_ x { x  |  ps }
31, 2dfss2f 3001 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  ->  x  e.  { x  |  ps } ) )
4 abid 2071 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2071 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5imbi12i 237 . . 3  |-  ( ( x  e.  { x  |  ph }  ->  x  e.  { x  |  ps } )  <->  ( ph  ->  ps ) )
76albii 1400 . 2  |-  ( A. x ( x  e. 
{ x  |  ph }  ->  x  e.  {
x  |  ps }
)  <->  A. x ( ph  ->  ps ) )
83, 7bitri 182 1  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    e. wcel 1434   {cab 2069    C_ wss 2984
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2990  df-ss 2997
This theorem is referenced by:  abss  3074  ssab  3075  ss2abi  3077  ss2abdv  3078  ss2rab  3081  rabss2  3088  iotanul  4947  iotass  4949
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