Theorem csbco 2980

Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)

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 2972	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2225	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 2936	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		sbcco 2899	. . . 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 2230	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
7		df-csb 2972	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
8		df-csb 2972	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
9	6, 7, 8	3eqtr4i 2145	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   = wceq 1314   e. wcel 1463   {cab 2101   [.wsbc 2878   [_csb 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-csb 2972

This theorem is referenced by:  csbvarg  2996  csbnest1g  3021  zsumdc  11042  fsum3  11045  fsumsplitf  11066