Theorem abeq2i 2300

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 2256	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2177	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2160   {cab 2175

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185

This theorem is referenced by:  rabid  2666  vex  2755  csbco  3082  csbcow  3083  csbnestgf  3124  ifmdc  3589  pwss  3606  snsspw  3779  iunpw  4498  ordon  4503  funcnv3  5297  tfrlem4  6339  tfrlem8  6344  tfrlem9  6345  tfrlemibxssdm  6353  tfr1onlembxssdm  6369  tfrcllembxssdm  6382  ixpm  6757  mapsnen  6838  sbthlem1  6987  1idprl  7620  1idpru  7621  recexprlem1ssl  7663  recexprlem1ssu  7664  recexprlemss1l  7665  recexprlemss1u  7666  txbas  14235