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Theorem abeq1i 2558
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
Hypothesis
Ref Expression
abeqri.1  |-  { x  |  ph }  =  A
Assertion
Ref Expression
abeq1i  |-  ( ph  <->  x  e.  A )

Proof of Theorem abeq1i
StepHypRef Expression
1 abeqri.1 . . . 4  |-  { x  |  ph }  =  A
21eqcomi 2442 . . 3  |-  A  =  { x  |  ph }
32abeq2i 2556 . 2  |-  ( x  e.  A  <->  ph )
43bicomi 205 1  |-  ( ph  <->  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1870   {cab 2414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424
This theorem is referenced by: (None)
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