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Theorem abeq1 2402
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 2401 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 eqcom 2298 . 2  |-  ( { x  |  ph }  =  A  <->  A  =  {
x  |  ph }
)
3 bicom 191 . . 3  |-  ( (
ph 
<->  x  e.  A )  <-> 
( x  e.  A  <->  ph ) )
43albii 1556 . 2  |-  ( A. x ( ph  <->  x  e.  A )  <->  A. x
( x  e.  A  <->  ph ) )
51, 2, 43bitr4i 268 1  |-  ( { x  |  ph }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282
This theorem is referenced by:  abbi1dv  2412  disj  3508  euabsn2  3711  dm0rn0  4911  dffo3  5691  dfsup2  7211  dfsup2OLD  7212  rankf  7482  dfon3  24503  dfiota3  24533  inpc  25380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292
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