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Theorem abeq1i 2404
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 2284 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 abeqri.1 . . 3  |-  { x  |  ph }  =  A
32eleq2i 2360 . 2  |-  ( x  e.  { x  | 
ph }  <->  x  e.  A )
41, 3bitr3i 242 1  |-  ( ph  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292
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