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Theorem ss2ab 3531
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 2618 . . 3  |-  F/_ x { x  |  ph }
2 nfab1 2618 . . 3  |-  F/_ x { x  |  ps }
31, 2dfss2f 3458 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  ->  x  e.  { x  |  ps } ) )
4 abid 2441 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2441 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5imbi12i 326 . . 3  |-  ( ( x  e.  { x  |  ph }  ->  x  e.  { x  |  ps } )  <->  ( ph  ->  ps ) )
76albii 1611 . 2  |-  ( A. x ( x  e. 
{ x  |  ph }  ->  x  e.  {
x  |  ps }
)  <->  A. x ( ph  ->  ps ) )
83, 7bitri 249 1  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    e. wcel 1758   {cab 2439    C_ wss 3439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-in 3446  df-ss 3453
This theorem is referenced by:  abss  3532  ssab  3533  ss2abi  3535  ss2abdv  3536  ss2rab  3539  rabss2  3546  rabsssn  30889
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