Theorem csbid 3053

Description: Analog of sbid 1762 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 3046	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2966	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2282	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2287	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2190	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1343   e. wcel 2136   {cab 2151   [.wsbc 2951   [_csb 3045

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbeq1a  3054  fvmpt2  5569  fsumsplitf  11349  ctiunctlemfo  12372