Theorem csbid 3438

Description: Analog of sbid 1997 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 3431	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3344	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2591	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2597	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2490	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1395   e. wcel 1819   {cab 2442   [.wsbc 3327   [_csb 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-sbc 3328  df-csb 3431

This theorem is referenced by:  csbeq1a  3439  fvmpt2i  5963  fvmpt2curryd  7018  gsummoncoe1  18473  gsumply1eq  18474  disji2f  27577  disjif2  27581  disjabrex  27582  disjabrexf  27583  fvmpt2f  27647  gsummpt2co  27931  measiuns  28361  fphpd  30954  fsumsplitf  31771  dvmptmulf  31937