Theorem csbco 3090

Description: Composition law for chained substitutions into a class.

Use the weaker csbcow 3091 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
csbco	|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Distinct variable group:   y, B

Allowed substitution hints:   A( x, y)   B( x)

Proof of Theorem csbco

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3081	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2304	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 3045	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		sbcco 3007	. . . 4 |- ( [. A / y ]. [. y / x ]. z e. B <-> [. A / x ]. z e. B )
5	3, 4	bitri 184	. . 3 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / x ]. z e. B )
6	5	abbii 2309	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
7		df-csb 3081	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
8		df-csb 3081	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
9	6, 7, 8	3eqtr4i 2224	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   = wceq 1364   e. wcel 2164   {cab 2179   [.wsbc 2985   [_csb 3080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbvarg  3108  csbnest1g  3136  zsumdc  11527  fsum3  11530  fsumsplitf  11551