Theorem abid 2165

Description: Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
abid	|- ( x e. { x | ph } <-> ph )

Proof of Theorem abid

Step	Hyp	Ref	Expression
1		df-clab 2164	. 2 |- ( x e. { x | ph } <-> [ x / x ] ph )
2		sbid 1774	. 2 |- ( [ x / x ] ph <-> ph )
3	1, 2	bitri 184	1 |- ( x e. { x | ph } <-> ph )

Syntax hints:   <-> wb 105   [wsb 1762   e. wcel 2148   {cab 2163

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-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164

This theorem is referenced by:  abeq2  2286  abeq2i  2288  abeq1i  2289  abeq2d  2290  abid2f  2345  elabgt  2880  elabgf  2881  ralab2  2903  rexab2  2905  sbccsbg  3088  sbccsb2g  3089  ss2ab  3225  abn0r  3449  abn0m  3450  tpid3g  3709  eluniab  3823  elintab  3857  iunab  3935  iinab  3950  intexabim  4154  iinexgm  4156  opm  4236  finds2  4602  dmmrnm  4848  sniota  5209  eusvobj2  5863  eloprabga  5964  indpi  7343  elabgf0  14568