Theorem csbid 2859

Description: Analog of sbid 1657 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 2853	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 2779	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2153	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2158	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2064	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1243   e. wcel 1393   {cab 2026   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbeq1a  2860  fvmpt2  5254