Theorem abeq2i 2316

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 2272	. 2 |- ( x e. A <-> x e. { x | ph } )
3		abid 2193	. 2 |- ( x e. { x | ph } <-> ph )
4	2, 3	bitri 184	1 |- ( x e. A <-> ph )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2176   {cab 2191

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201

This theorem is referenced by:  rabid  2682  vex  2775  csbco  3103  csbcow  3104  csbnestgf  3146  ifmdc  3612  pwss  3632  snsspw  3805  iunpw  4527  ordon  4534  funcnv3  5336  tfrlem4  6399  tfrlem8  6404  tfrlem9  6405  tfrlemibxssdm  6413  tfr1onlembxssdm  6429  tfrcllembxssdm  6442  ixpm  6817  mapsnen  6903  sbthlem1  7059  1idprl  7703  1idpru  7704  recexprlem1ssl  7746  recexprlem1ssu  7747  recexprlemss1l  7748  recexprlemss1u  7749  txbas  14730