Theorem csbid 3541

Description: Analogue of sbid 2114 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 3534	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3452	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2739	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2745	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2648	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbeq1a  3542  fvmpt2f  6283  fvmpt2i  6290  fvmpt2curryd  7397  fsumsplitf  14472  gsummoncoe1  19674  gsumply1eq  19675  disji2f  29390  disjif2  29394  disjabrex  29395  disjabrexf  29396  gsummpt2co  29780  measiuns  30280  fphpd  37380  disjrnmpt2  39375  climinf2mpt  39946  climinfmpt  39947  dvmptmulf  40152  sge0f1o  40599