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Theorem abeq1i 2538
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 2418 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 abeqri.1 . . 3  |-  { x  |  ph }  =  A
32eleq2i 2494 . 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 1652    e. wcel 1725   {cab 2416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426
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