ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abeq2d Unicode version

Theorem abeq2d 2166
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeqd.1  |-  ( ph  ->  A  =  { x  |  ps } )
Assertion
Ref Expression
abeq2d  |-  ( ph  ->  ( x  e.  A  <->  ps ) )

Proof of Theorem abeq2d
StepHypRef Expression
1 abeqd.1 . . 3  |-  ( ph  ->  A  =  { x  |  ps } )
21eleq2d 2123 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2044 . 2  |-  ( x  e.  { x  |  ps }  <->  ps )
42, 3syl6bb 189 1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by:  fvelimab  5257  frecsuclem3  6021
  Copyright terms: Public domain W3C validator