Theorem csbid 3155

Description: Analog of sbid 1827 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 3148	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3067	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2354	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2361	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2263	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1402   e. wcel 2209   {cab 2224   [.wsbc 3051   [_csb 3147

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is referenced by:  csbeq1a  3156  fvmpt2  5783  fsumsplitf  12153  ctiunctlemfo  13308