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Theorem abeq1i 2551
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
Hypothesis
Ref Expression
abeqri.1  |-  { x  |  ph }  =  A
Assertion
Ref Expression
abeq1i  |-  ( ph  <->  x  e.  A )

Proof of Theorem abeq1i
StepHypRef Expression
1 abid 2431 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 abeqri.1 . . 3  |-  { x  |  ph }  =  A
32eleq2i 2507 . 2  |-  ( x  e.  { x  | 
ph }  <->  x  e.  A )
41, 3bitr3i 244 1  |-  ( ph  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1654    e. wcel 1728   {cab 2429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-11 1764  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439
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