Theorem abeq2i 2190

Description: Equality of a class variable and a class abstraction (inference rule). (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 2146	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2070	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 182	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078

This theorem is referenced by:  rabid  2530  vex  2605  csbco  2918  csbnestgf  2955  pwss  3405  snsspw  3564  iunpw  4237  ordon  4238  funcnv3  4992  tfrlem4  5962  tfrlem8  5967  tfrlem9  5968  tfrlemibxssdm  5976  tfr1onlembxssdm  5992  tfrcllembxssdm  6005  1idprl  6842  1idpru  6843  recexprlem1ssl  6885  recexprlem1ssu  6886  recexprlemss1l  6887  recexprlemss1u  6888