Theorem csbid 3057

Description: Analog of sbid 1767 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 3050	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2970	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2286	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2291	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2195	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1348   e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbeq1a  3058  fvmpt2  5579  fsumsplitf  11371  ctiunctlemfo  12394