Theorem csbid 3391

Description: Analog of sbid 1949 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 3384	. 2 |- [_ x / x ]_ A = { y | [. x / x ]. y e. A }
2		sbcid 3298	. . 3 |- ( [. x / x ]. y e. A <-> y e. A )
3	2	abbii 2583	. 2 |- { y | [. x / x ]. y e. A } = { y | y e. A }
4		abid2 2589	. 2 |- { y | y e. A } = A
5	1, 3, 4	3eqtri 2483	1 |- [_ x / x ]_ A = A

Syntax hints:   = wceq 1370   e. wcel 1758   {cab 2436   [.wsbc 3281   [_csb 3383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-sbc 3282  df-csb 3384

This theorem is referenced by:  csbeq1a  3392  fvmpt2i  5876  fvmpt2curryd  6887  disji2f  26052  disjif2  26056  disjabrex  26057  disjabrexf  26058  fvmpt2f  26106  gsummpt2co  26380  measiuns  26762  fphpd  29290  gsummoncoe1  30982  gsumply1eq  30991