Theorem csbcog 2007

Description: Composition law for chained substitutions into a class.

Assertion

Ref	Expression
csbcog	|- (A e. C -> [_A / y]_[_y / x]_B = [_A / x]_B)

Distinct variable group:   y,B

Proof of Theorem csbcog

Step	Hyp	Ref	Expression
1		df-csb 2002	. . . . . 6 |- [_y / x]_B = {z | [y / x]z e. B}
2	1	abeq2i 1570	. . . . 5 |- (z e. [_y / x]_B <-> [y / x]z e. B)
3	2	sbcbii 1978	. . . 4 |- (A e. C -> ([A / y]z e. [_y / x]_B <-> [A / y][y / x]z e. B))
4		sbccog 1952	. . . 4 |- (A e. C -> ([A / y][y / x]z e. B <-> [A / x]z e. B))
5	3, 4	bitrd 528	. . 3 |- (A e. C -> ([A / y]z e. [_y / x]_B <-> [A / x]z e. B))
6	5	abbidv 1577	. 2 |- (A e. C -> {z | [A / y]z e. [_y / x]_B} = {z | [A / x]z e. B})
7		df-csb 2002	. 2 |- [_A / y]_[_y / x]_B = {z | [A / y]z e. [_y / x]_B}
8		df-csb 2002	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
9	6, 7, 8	3eqtr4g 1531	1 |- (A e. C -> [_A / y]_[_y / x]_B = [_A / x]_B)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  [_csb 2001

This theorem is referenced by:  csbvarg 2021  csbnestg 2036  sbcbrg 2662  csbima12g 3413  csbfv12g 3742  csboprg 3986  eqerlem 4270  csbnegg 5364

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

Copyright terms: Public domain