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Theorem abeq2i 2594
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
Hypothesis
Ref Expression
abeqi.1  |-  A  =  { x  |  ph }
Assertion
Ref Expression
abeq2i  |-  ( x  e.  A  <->  ph )

Proof of Theorem abeq2i
StepHypRef Expression
1 abeqi.1 . . . 4  |-  A  =  { x  |  ph }
21a1i 11 . . 3  |-  ( T. 
->  A  =  {
x  |  ph }
)
32abeq2d 2593 . 2  |-  ( T. 
->  ( x  e.  A  <->  ph ) )
43trud 1388 1  |-  ( x  e.  A  <->  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   T. wtru 1380    e. wcel 1767   {cab 2452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462
This theorem is referenced by:  abeq1i  2596  rabid  3038  vex  3116  csbco  3445  csbnestgf  3840  elsn  4041  funcnv3  5647  opabiota  5928  zfrep6  6749  tfrlem4  7045  tfrlem8  7050  tfrlem9  7051  ixpn0  7498  mapsnen  7590  sbthlem1  7624  dffi3  7887  1idpr  9403  ltexprlem1  9410  ltexprlem2  9411  ltexprlem3  9412  ltexprlem4  9413  ltexprlem6  9415  ltexprlem7  9416  reclem2pr  9422  reclem3pr  9423  reclem4pr  9424  supsrlem  9484  txbas  19803  xkoccn  19855  xkoptsub  19890  xkoco1cn  19893  xkoco2cn  19894  xkoinjcn  19923  mbfi1fseqlem4  21860  avril1  24847  rnmpt2ss  27187  setinds  28787  wfrlem2  28921  wfrlem3  28922  wfrlem4  28923  wfrlem9  28928  frrlem2  28965  frrlem3  28966  frrlem4  28967  frrlem5e  28972  frrlem11  28976  sdclem1  29839  csbcom2fi  30138  csbgfi  30139  bnj1436  32977  bnj916  33070  bnj983  33088  bnj1083  33113  bnj1245  33149  bnj1311  33159  bnj1371  33164  bnj1398  33169  bj-ififc  33247  bj-elsngl  33607  bj-projun  33633  bj-projval  33635
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