Theorem csbid 2915

Description: Analog of sbid 1697 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 2909	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2830	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2194	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2199	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2105	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbeq1a  2916  fvmpt2  5275