Theorem csbid 3088

Description: Analog of sbid 1785 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 3081	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3001	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2309	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2314	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2218	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1364   e. wcel 2164   {cab 2179   [.wsbc 2985   [_csb 3080

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbeq1a  3089  fvmpt2  5641  fsumsplitf  11551  ctiunctlemfo  12596