Theorem csbid 2008

Description: Analog of sbid 1186 for proper substitution into a class.

Assertion

Ref	Expression
csbid	|- [_x / x]_A = A

Proof of Theorem csbid

Step	Hyp	Ref	Expression
1		df-csb 2005	. 2 |- [_x / x]_A = {y | [x / x]y e. A}
2		sbid 1186	. . 3 |- ([x / x]y e. A <-> y e. A)
3	2	abbii 1578	. 2 |- {y | [x / x]y e. A} = {y | y e. A}
4		abid2 1583	. 2 |- {y | y e. A} = A
5	1, 3, 4	3eqtr 1502	1 |- [_x / x]_A = A

Syntax hints:   = wceq 958   e. wcel 960  [wsbc 1172  {cab 1466  [_csb 2004

This theorem is referenced by:  csbeq1a 2009

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-csb 2005

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