Theorem csbid 3101

Description: Analog of sbid 1797 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 3094	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3014	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2321	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2326	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2230	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   [.wsbc 2998   [_csb 3093

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999  df-csb 3094

This theorem is referenced by:  csbeq1a  3102  fvmpt2  5663  fsumsplitf  11719  ctiunctlemfo  12810