Theorem csbid 3105

Description: Analog of sbid 1798 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 3098	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3018	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2322	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2327	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2231	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1373   e. wcel 2177   {cab 2192   [.wsbc 3002   [_csb 3097

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-sbc 3003  df-csb 3098

This theorem is referenced by:  csbeq1a  3106  fvmpt2  5681  fsumsplitf  11804  ctiunctlemfo  12895