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Theorem abeq1 2527
Description: Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
abeq1  |-  ( { x  |  ph }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem abeq1
StepHypRef Expression
1 abeq2 2526 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 eqcom 2411 . 2  |-  ( { x  |  ph }  =  A  <->  A  =  {
x  |  ph }
)
3 bicom 200 . . 3  |-  ( (
ph 
<->  x  e.  A )  <-> 
( x  e.  A  <->  ph ) )
43albii 1661 . 2  |-  ( A. x ( ph  <->  x  e.  A )  <->  A. x
( x  e.  A  <->  ph ) )
51, 2, 43bitr4i 277 1  |-  ( { x  |  ph }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1403    = wceq 1405    e. wcel 1842   {cab 2387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397
This theorem is referenced by:  abbi1dvOLD  2541  disj  3810  euabsn2  4043  dm0rn0  5040  dffo3  6024  dfsup2  7936  dfsup2OLD  7937  rankf  8244  dfon3  30230  dfiota3  30261
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