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Theorem abeq1 2548
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 2547 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 eqcom 2432 . 2  |-  ( { x  |  ph }  =  A  <->  A  =  {
x  |  ph }
)
3 bicom 204 . . 3  |-  ( (
ph 
<->  x  e.  A )  <-> 
( x  e.  A  <->  ph ) )
43albii 1688 . 2  |-  ( A. x ( ph  <->  x  e.  A )  <->  A. x
( x  e.  A  <->  ph ) )
51, 2, 43bitr4i 281 1  |-  ( { x  |  ph }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1436    = wceq 1438    e. wcel 1869   {cab 2408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418
This theorem is referenced by:  abbi1dvOLD  2562  disj  3834  euabsn2  4069  dm0rn0  5068  dffo3  6050  dfsup2  7962  rankf  8268  dfon3  30658  dfiota3  30689  dffo3f  37344
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