Theorem cleqabi 1176

Description: Equality of a class variable and a class abstraction (inference rule).

Hypothesis

Ref	Expression
cleqabi.1	|- A = {x | ph}

Assertion

Ref	Expression
cleqabi	|- (x e. A <-> ph)

Proof of Theorem cleqabi

Step	Hyp	Ref	Expression
1		cleqabi.1	. . 3 |- A = {x | ph}
2	1	eleq2i 1153	. 2 |- (x e. A <-> x e. {x | ph})
3		abid 1094	. 2 |- (x e. {x | ph} <-> ph)
4	2, 3	bitr 151	1 |- (x e. A <-> ph)

Syntax hints:   <-> wb 127  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  rabid 1308  visset 1350  noel 1711  elpw 1801  elsn 1820  snsspw 1857  pw0 1882  iunpw 2040  elopab 2110  dmco2 2673  zfrep6 2744  fv2 2828  tz6.12-1 2842  fopab2 2891  tfrlem4 2952  tfrlem5 2953  tfrlem8 2956  tfrlem9 2957  mapsnen 3334  sbthlem1 3349  ac6lem 3575  1pr 3911  1idpr 3927  ltexprlem1 3936  ltexprlem2 3937  ltexprlem3 3938  ltexprlem4 3939  ltexprlem6 3941  ltexprlem7 3942  reclem1pr 3950  reclem2pr 3951  reclem3pr 3952  reclem4pr 3953  suppsrlem 4015  suppsr3 4018  suprelem 4053  chsscm 5147  chcmh 5148

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org