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Theorem ss2ab 3508
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 2604 . . 3  |-  F/_ x { x  |  ph }
2 nfab1 2604 . . 3  |-  F/_ x { x  |  ps }
31, 2dfss2f 3434 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  ->  x  e.  { x  |  ps } ) )
4 abid 2449 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2449 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5imbi12i 332 . . 3  |-  ( ( x  e.  { x  |  ph }  ->  x  e.  { x  |  ps } )  <->  ( ph  ->  ps ) )
76albii 1701 . 2  |-  ( A. x ( x  e. 
{ x  |  ph }  ->  x  e.  {
x  |  ps }
)  <->  A. x ( ph  ->  ps ) )
83, 7bitri 257 1  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452    e. wcel 1897   {cab 2447    C_ wss 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-in 3422  df-ss 3429
This theorem is referenced by:  abss  3509  ssab  3510  ss2abi  3512  ss2abdv  3513  ss2rab  3516  rabss2  3523  rabsssn  4013  clss2lem  36262
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