Theorem csbcog 2003

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 1998	. . . . . 6 |- [_y / x]_B = {z | [y / x]z e. B}
2	1	abeq2i 1567	. . . . 5 |- (z e. [_y / x]_B <-> [y / x]z e. B)
3	2	sbcbii 1974	. . . 4 |- (A e. C -> ([A / y]z e. [_y / x]_B <-> [A / y][y / x]z e. B))
4		sbccog 1948	. . . 4 |- (A e. C -> ([A / y][y / x]z e. B <-> [A / x]z e. B))
5	3, 4	bitrd 527	. . 3 |- (A e. C -> ([A / y]z e. [_y / x]_B <-> [A / x]z e. B))
6	5	abbidv 1574	. 2 |- (A e. C -> {z | [A / y]z e. [_y / x]_B} = {z | [A / x]z e. B})
7		df-csb 1998	. 2 |- [_A / y]_[_y / x]_B = {z | [A / y]z e. [_y / x]_B}
8		df-csb 1998	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
9	6, 7, 8	3eqtr4g 1528	1 |- (A e. C -> [_A / y]_[_y / x]_B = [_A / x]_B)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  [wsbc 1168  {cab 1461  [_csb 1997

This theorem is referenced by:  csbvarg 2017  csbnestg 2032  sbcbrg 2657  csbima12g 3405  csbfv12g 3733  csboprg 3977  eqerlem 4260  csbnegg 5344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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