Theorem biabri 1180

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

Hypothesis

Ref	Expression
biabri.1	|- (x e. A <-> ph)

Assertion

Ref	Expression
biabri	|- A = {x | ph}

Distinct variable group(s):   x,A

Proof of Theorem biabri

Step	Hyp	Ref	Expression
1		cleqab 1174	. 2 |- (A = {x | ph} <-> A.x(x e. A <-> ph))
2		biabri.1	. 2 |- (x e. A <-> ph)
3	1, 2	mpgbir 686	1 |- A = {x | ph}

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

This theorem is referenced by:  abid2 1186  difeqri 1589  symdif2 1690  dfnul2 1709  dfpr2 1821  dftp2 1835  pw0 1882  iunrab 2022  fv3 2839  tfrlem3 2951  mapsn 3269  unfilem1 3438  dfom4 3479  cardnum 3694  alephiso 3697  nnz 4582  nn0z 4583  dfch2 5254  pjrn 5587

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  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

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