Theorem csbid 2941

Description: Analog of sbid 1705 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 2935	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2856	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2204	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2209	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2113	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1290   e. wcel 1439   {cab 2075   [.wsbc 2841   [_csb 2934

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-sbc 2842  df-csb 2935

This theorem is referenced by:  csbeq1a  2942  fvmpt2  5399  fsumsplitf  10856