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Theorem abss 3509
Description: Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
abss  |-  ( { x  |  ph }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem abss
StepHypRef Expression
1 abid2 2583 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq2i 3468 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  x  e.  A }  <->  { x  |  ph }  C_  A
)
3 ss2ab 3508 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  x  e.  A }  <->  A. x
( ph  ->  x  e.  A ) )
42, 3bitr3i 259 1  |-  ( { x  |  ph }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
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:  abssdv  3514  rabss  3517  uniiunlem  3528  iunss  4332  moabex  4672  reliun  4972  axdc2lem  8903  mptelee  24973  fpwrelmap  28366  ss2iundf  36295  hoidmvlelem1  38454
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