Theorem abeq2i 2250

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 2206	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2127	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 183	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135

This theorem is referenced by:  rabid  2606  vex  2689  csbco  3013  csbnestgf  3052  ifmdc  3509  pwss  3526  snsspw  3691  iunpw  4401  ordon  4402  funcnv3  5185  tfrlem4  6210  tfrlem8  6215  tfrlem9  6216  tfrlemibxssdm  6224  tfr1onlembxssdm  6240  tfrcllembxssdm  6253  ixpm  6624  mapsnen  6705  sbthlem1  6845  1idprl  7398  1idpru  7399  recexprlem1ssl  7441  recexprlem1ssu  7442  recexprlemss1l  7443  recexprlemss1u  7444  txbas  12427