Theorem csbid 3443

Description: Analog of sbid 1965 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 3436	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3348	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2601	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2607	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2500	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1379   e. wcel 1767   {cab 2452   [.wsbc 3331   [_csb 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-sbc 3332  df-csb 3436

This theorem is referenced by:  csbeq1a  3444  fvmpt2i  5954  fvmpt2curryd  6997  gsummoncoe1  18114  gsumply1eq  18115  disji2f  27108  disjif2  27112  disjabrex  27113  disjabrexf  27114  fvmpt2f  27167  gsummpt2co  27431  measiuns  27825  fphpd  30352