Theorem csbid 3092

Description: Analog of sbid 1788 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 3085	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3005	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2312	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2317	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2221	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1364   e. wcel 2167   {cab 2182   [.wsbc 2989   [_csb 3084

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990  df-csb 3085

This theorem is referenced by:  csbeq1a  3093  fvmpt2  5645  fsumsplitf  11573  ctiunctlemfo  12656