Theorem csbco 3148

Description: Composition law for chained substitutions into a class.

Use the weaker csbcow 3149 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 3139	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2343	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 3102	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		sbcco 3064	. . . 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 2348	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
7		df-csb 3139	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
8		df-csb 3139	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
9	6, 7, 8	3eqtr4i 2263	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   = wceq 1398   e. wcel 2203   {cab 2218   [.wsbc 3042   [_csb 3138

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sbc 3043  df-csb 3139

This theorem is referenced by:  csbvarg  3166  csbnest1g  3194  zsumdc  12070  fsum3  12073  fsumsplitf  12094