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Theorem abss 3404
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 2552 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq2i 3365 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  x  e.  A }  <->  { x  |  ph }  C_  A
)
3 ss2ab 3403 . 2  |-  ( { x  |  ph }  C_ 
{ x  |  x  e.  A }  <->  A. x
( ph  ->  x  e.  A ) )
42, 3bitr3i 243 1  |-  ( { x  |  ph }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725   {cab 2421    C_ wss 3312
This theorem is referenced by:  abssdv  3409  rabss  3412  uniiunlem  3423  iunss  4124  moabex  4414  reliun  4987  axdc2lem  8320  mptelee  25826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326
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