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Theorem ss2ab 3504
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 2564 . . 3  |-  F/_ x { x  |  ph }
2 nfab1 2564 . . 3  |-  F/_ x { x  |  ps }
31, 2dfss2f 3430 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  ->  x  e.  { x  |  ps } ) )
4 abid 2387 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2387 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5imbi12i 324 . . 3  |-  ( ( x  e.  { x  |  ph }  ->  x  e.  { x  |  ps } )  <->  ( ph  ->  ps ) )
76albii 1659 . 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 1401    e. wcel 1840   {cab 2385    C_ wss 3411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-in 3418  df-ss 3425
This theorem is referenced by:  abss  3505  ssab  3506  ss2abi  3508  ss2abdv  3509  ss2rab  3512  rabss2  3519  rabsssn  38364
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