Theorem csbnestg 2036

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
csbnestg	|- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Distinct variable groups:   x,C   x,y

Proof of Theorem csbnestg

Step	Hyp	Ref	Expression
1		csbcog 2007	. . . . 5 |- (A e. V -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / z]_[_z / y]_C)
2	1	adantr 389	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / z]_[_z / y]_C)
3		visset 1813	. . . . . . . 8 |- w e. V
4		csbnestglem 2035	. . . . . . . 8 |- ((w e. V /\ A.x B e. V) -> [_w / x]_[_B / z]_[_z / y]_C = [_[_w / x]_B / z]_[_z / y]_C)
5	3, 4	mpan 695	. . . . . . 7 |- (A.x B e. V -> [_w / x]_[_B / z]_[_z / y]_C = [_[_w / x]_B / z]_[_z / y]_C)
6	5	csbeq2dv 2019	. . . . . 6 |- ((A.x B e. V /\ A e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / w]_[_[_w / x]_B / z]_[_z / y]_C)
7	6	ancoms 436	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / w]_[_[_w / x]_B / z]_[_z / y]_C)
8		csbnestglem 2035	. . . . . 6 |- ((A e. V /\ A.w[_w / x]_B e. V) -> [_A / w]_[_[_w / x]_B / z]_[_z / y]_C = [_[_A / w]_[_w / x]_B / z]_[_z / y]_C)
9		csbexg 2008	. . . . . . . 8 |- ((w e. V /\ A.x B e. V) -> [_w / x]_B e. V)
10	3, 9	mpan 695	. . . . . . 7 |- (A.x B e. V -> [_w / x]_B e. V)
11	10	19.21aiv 1286	. . . . . 6 |- (A.x B e. V -> A.w[_w / x]_B e. V)
12	8, 11	sylan2 451	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_[_w / x]_B / z]_[_z / y]_C = [_[_A / w]_[_w / x]_B / z]_[_z / y]_C)
13		csbcog 2007	. . . . . . 7 |- (A e. V -> [_A / w]_[_w / x]_B = [_A / x]_B)
14	13	csbeq1d 2004	. . . . . 6 |- (A e. V -> [_[_A / w]_[_w / x]_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
15	14	adantr 389	. . . . 5 |- ((A e. V /\ A.x B e. V) -> [_[_A / w]_[_w / x]_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
16	7, 12, 15	3eqtrd 1511	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
17	2, 16	eqtr3d 1509	. . 3 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
18		hba1 1003	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
19		csbcog 2007	. . . . . 6 |- (B e. V -> [_B / z]_[_z / y]_C = [_B / y]_C)
20	19	a4s 984	. . . . 5 |- (A.x B e. V -> [_B / z]_[_z / y]_C = [_B / y]_C)
21	18, 20	csbeq2d 2018	. . . 4 |- ((A.x B e. V /\ A e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
22	21	ancoms 436	. . 3 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
23		csbexg 2008	. . . 4 |- ((A e. V /\ A.x B e. V) -> [_A / x]_B e. V)
24		csbcog 2007	. . . 4 |- ([_A / x]_B e. V -> [_[_A / x]_B / z]_[_z / y]_C = [_[_A / x]_B / y]_C)
25	23, 24	syl 10	. . 3 |- ((A e. V /\ A.x B e. V) -> [_[_A / x]_B / z]_[_z / y]_C = [_[_A / x]_B / y]_C)
26	17, 22, 25	3eqtr3d 1515	. 2 |- ((A e. V /\ A.x B e. V) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
27		elisset 1817	. 2 |- (A e. R -> A e. V)
28		elisset 1817	. . 3 |- (B e. S -> B e. V)
29	28	19.20i 992	. 2 |- (A.x B e. S -> A.x B e. V)
30	26, 27, 29	syl2an 454	1 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811  [_csb 2001

This theorem is referenced by:  sbcnestg 2038  csbco3g 2040

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-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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