Theorem abeq2i 2318

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 2274	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2195	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203

This theorem is referenced by:  rabid  2684  vex  2779  csbco  3111  csbcow  3112  csbnestgf  3154  ifmdc  3622  pwss  3642  snsspw  3818  iunpw  4545  ordon  4552  funcnv3  5355  tfrlem4  6422  tfrlem8  6427  tfrlem9  6428  tfrlemibxssdm  6436  tfr1onlembxssdm  6452  tfrcllembxssdm  6465  ixpm  6840  mapsnen  6927  sbthlem1  7085  1idprl  7738  1idpru  7739  recexprlem1ssl  7781  recexprlem1ssu  7782  recexprlemss1l  7783  recexprlemss1u  7784  txbas  14845