Theorem csbid 3100

Description: Analog of sbid 1796 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 3093	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3013	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2320	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2325	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2229	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1372   e. wcel 2175   {cab 2190   [.wsbc 2997   [_csb 3092

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-sbc 2998  df-csb 3093

This theorem is referenced by:  csbeq1a  3101  fvmpt2  5662  fsumsplitf  11690  ctiunctlemfo  12781