Theorem abeq2i 2345

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 2301	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2222	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230

This theorem is referenced by:  rabid  2721  vex  2818  csbco  3150  csbcow  3151  csbnestgf  3193  ifmdc  3667  pwss  3690  snsspw  3870  iunpw  4603  ordon  4610  funcnv3  5420  tfrlem4  6546  tfrlem8  6551  tfrlem9  6552  tfrlemibxssdm  6560  tfr1onlembxssdm  6576  tfrcllembxssdm  6589  ixpm  6967  mapsnen  7055  sbthlem1  7229  1idprl  7907  1idpru  7908  recexprlem1ssl  7950  recexprlem1ssu  7951  recexprlemss1l  7952  recexprlemss1u  7953  txbas  15140