Theorem csbco 3543

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 3534	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2735	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 3491	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		sbcco 3458	. . . 4 |- ( [. A / y ]. [. y / x ]. z e. B <-> [. A / x ]. z e. B )
5	3, 4	bitri 264	. . 3 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / x ]. z e. B )
6	5	abbii 2739	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
7		df-csb 3534	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
8		df-csb 3534	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
9	6, 7, 8	3eqtr4i 2654	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbnest1g  4001  csbvarg  4003  fvmpt2curryd  7397  zsum  14449  fsum  14451  fsumsplitf  14472  zprod  14667  fprod  14671  gsumply1eq  19675  bj-csbsn  32899  sbccom2  33930  disjrnmpt2  39375  disjinfi  39380  climinf2mpt  39946  climinfmpt  39947  dvmptmulf  40152  dvmptfprod  40160