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Theorem abeq1 2494
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 2493 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 eqcom 2390 . 2  |-  ( { x  |  ph }  =  A  <->  A  =  {
x  |  ph }
)
3 bicom 192 . . 3  |-  ( (
ph 
<->  x  e.  A )  <-> 
( x  e.  A  <->  ph ) )
43albii 1572 . 2  |-  ( A. x ( ph  <->  x  e.  A )  <->  A. x
( x  e.  A  <->  ph ) )
51, 2, 43bitr4i 269 1  |-  ( { x  |  ph }  =  A  <->  A. x ( ph  <->  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2374
This theorem is referenced by:  abbi1dv  2504  disj  3612  euabsn2  3819  dm0rn0  5027  dffo3  5824  dfsup2  7383  dfsup2OLD  7384  rankf  7654  dfon3  25457  dfiota3  25487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384
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