Theorem csbid 3132

Description: Analog of sbid 1820 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbid	|- [_ x / x ]_ A = A

Proof of Theorem csbid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3125	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3044	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2345	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2350	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2254	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1395   e. wcel 2200   {cab 2215   [.wsbc 3028   [_csb 3124

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbeq1a  3133  fvmpt2  5717  fsumsplitf  11914  ctiunctlemfo  13005