Theorem abeq2i 2223

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 2179	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2101	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 183	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 104   = wceq 1312   e. wcel 1461   {cab 2099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109

This theorem is referenced by:  rabid  2578  vex  2658  csbco  2978  csbnestgf  3016  ifmdc  3473  pwss  3490  snsspw  3655  iunpw  4359  ordon  4360  funcnv3  5141  tfrlem4  6162  tfrlem8  6167  tfrlem9  6168  tfrlemibxssdm  6176  tfr1onlembxssdm  6192  tfrcllembxssdm  6205  ixpm  6576  mapsnen  6657  sbthlem1  6795  1idprl  7340  1idpru  7341  recexprlem1ssl  7383  recexprlem1ssu  7384  recexprlemss1l  7385  recexprlemss1u  7386  txbas  12263