Theorem csbid 3006

Description: Analog of sbid 1747 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 2999	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2919	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2253	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2258	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2162	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1331   e. wcel 1480   {cab 2123   [.wsbc 2904   [_csb 2998

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905  df-csb 2999

This theorem is referenced by:  csbeq1a  3007  fvmpt2  5497  fsumsplitf  11170  ctiunctlemfo  11941