Theorem abeq2i 2349

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 2305	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2226	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  rabid  2727  vex  2824  csbco  3157  csbcow  3158  csbnestgf  3200  ifmdc  3683  pwss  3707  snsspw  3887  iunpw  4624  ordon  4631  funcnv3  5441  tfrlem4  6577  tfrlem8  6582  tfrlem9  6583  tfrlemibxssdm  6591  tfr1onlembxssdm  6607  tfrcllembxssdm  6620  ixpm  7005  mapsnen  7093  sbthlem1  7267  1idprl  7950  1idpru  7951  recexprlem1ssl  7993  recexprlem1ssu  7994  recexprlemss1l  7995  recexprlemss1u  7996  txbas  15285