Theorem csbid 3067

Description: Analog of sbid 1774 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 3060	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2980	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2293	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2298	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2202	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163   [.wsbc 2964   [_csb 3059

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2965  df-csb 3060

This theorem is referenced by:  csbeq1a  3068  fvmpt2  5601  fsumsplitf  11418  ctiunctlemfo  12442