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Theorem abeq1 2552
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 2551 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 eqcom 2445 . 2  |-  ( { x  |  ph }  =  A  <->  A  =  {
x  |  ph }
)
3 bicom 200 . . 3  |-  ( (
ph 
<->  x  e.  A )  <-> 
( x  e.  A  <->  ph ) )
43albii 1610 . 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 1367    = wceq 1369    e. wcel 1756   {cab 2429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439
This theorem is referenced by:  abbi1dvOLD  2566  disj  3738  euabsn2  3965  dm0rn0  5075  dffo3  5877  dfsup2  7711  dfsup2OLD  7712  rankf  8020  dfon3  27942  dfiota3  27973
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