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Theorem ss2ab 3254
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 2434 . . 3  |-  F/_ x { x  |  ph }
2 nfab1 2434 . . 3  |-  F/_ x { x  |  ps }
31, 2dfss2f 3184 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  ->  x  e.  { x  |  ps } ) )
4 abid 2284 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2284 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5imbi12i 316 . . 3  |-  ( ( x  e.  { x  |  ph }  ->  x  e.  { x  |  ps } )  <->  ( ph  ->  ps ) )
76albii 1556 . 2  |-  ( A. x ( x  e. 
{ x  |  ph }  ->  x  e.  {
x  |  ps }
)  <->  A. x ( ph  ->  ps ) )
83, 7bitri 240 1  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    e. wcel 1696   {cab 2282    C_ wss 3165
This theorem is referenced by:  abss  3255  ssab  3256  ss2abi  3258  ss2abdv  3259  ss2rab  3262  rabss2  3269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179
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