Theorem abeq2i 2307

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)

Hypothesis

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

Assertion

Ref	Expression
abeq2i	|- ( x e. A <-> ph )

Proof of Theorem abeq2i

Step	Hyp	Ref	Expression
1		abeqi.1	. . 3 |- A = { x | ph }
2	1	eleq2i 2263	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2184	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by:  rabid  2673  vex  2766  csbco  3094  csbcow  3095  csbnestgf  3137  ifmdc  3602  pwss  3622  snsspw  3795  iunpw  4516  ordon  4523  funcnv3  5321  tfrlem4  6380  tfrlem8  6385  tfrlem9  6386  tfrlemibxssdm  6394  tfr1onlembxssdm  6410  tfrcllembxssdm  6423  ixpm  6798  mapsnen  6879  sbthlem1  7032  1idprl  7674  1idpru  7675  recexprlem1ssl  7717  recexprlem1ssu  7718  recexprlemss1l  7719  recexprlemss1u  7720  txbas  14578