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Theorem abeq1i 2488
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 2368 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 abeqri.1 . . 3  |-  { x  |  ph }  =  A
32eleq2i 2444 . 2  |-  ( x  e.  { x  | 
ph }  <->  x  e.  A )
41, 3bitr3i 243 1  |-  ( ph  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   {cab 2366
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-11 1753  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376
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