Theorem csbco 3055

Description: Composition law for chained substitutions into a class.

Use the weaker csbcow 3056 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 3046	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2277	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 3010	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		sbcco 2972	. . . 4 |- ( [. A / y ]. [. y / x ]. z e. B <-> [. A / x ]. z e. B )
5	3, 4	bitri 183	. . 3 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / x ]. z e. B )
6	5	abbii 2282	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
7		df-csb 3046	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
8		df-csb 3046	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
9	6, 7, 8	3eqtr4i 2196	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   = wceq 1343   e. wcel 2136   {cab 2151   [.wsbc 2951   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbvarg  3073  csbnest1g  3100  zsumdc  11325  fsum3  11328  fsumsplitf  11349