Theorem csbco 3444

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

Assertion

Ref	Expression
csbco

Distinct variable group:   ,

Proof of Theorem csbco

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3435	. . . . . 6
2	1	abeq2i 2584	. . . . 5
3	2	sbcbii 3387	. . . 4
4		sbcco 3350	. . . 4
5	3, 4	bitri 249	. . 3
6	5	abbii 2591	. 2
7		df-csb 3435	. 2
8		df-csb 3435	. 2
9	6, 7, 8	3eqtr4i 2496	1

Syntax hints:  =wceq 1395  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434

This theorem is referenced by:  csbnest1g  3845  csbvarg  3848  fvmpt2curryd  7019  zsum  13540  fsum  13542  zprod  13744  fprod  13748  gsumply1eq  18347  sbccom2  30530  fsumsplitf  31568  dvmptmulf  31734  dvmptfprod  31742  bj-csbsn  34471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328  df-csb 3435