Theorem csbid 3135

Description: Analog of sbid 1822 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 3128	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3047	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2347	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2352	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2256	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1397   e. wcel 2202   {cab 2217   [.wsbc 3031   [_csb 3127

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032  df-csb 3128

This theorem is referenced by:  csbeq1a  3136  fvmpt2  5730  fsumsplitf  11968  ctiunctlemfo  13059