Theorem csbnestg 2581

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 2547	. . . . 5 |- (A e. _V -> [_A / w]_[_w / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / z]_[_z / y]_C)
2	1	adantr 425	. . . 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 2295	. . . . . . . 8 |- w e. _V
4		csbnestglem 2580	. . . . . . . 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 759	. . . . . . 7 |- (A.x B e. _V -> [_w / x]_[_B / z]_[_z / y]_C = [_[_w / x]_B / z]_[_z / y]_C)
6	5	csbeq2dv 2562	. . . . . 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 484	. . . . 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 2580	. . . . . 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 2548	. . . . . . . 8 |- ((w e. _V /\ A.x B e. _V) -> [_w / x]_B e. _V)
10	3, 9	mpan 759	. . . . . . 7 |- (A.x B e. _V -> [_w / x]_B e. _V)
11	10	19.21aiv 1664	. . . . . 6 |- (A.x B e. _V -> A.w[_w / x]_B e. _V)
12	8, 11	sylan2 500	. . . . 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 2547	. . . . . . 7 |- (A e. _V -> [_A / w]_[_w / x]_B = [_A / x]_B)
14	13	csbeq1d 2544	. . . . . 6 |- (A e. _V -> [_[_A / w]_[_w / x]_B / z]_[_z / y]_C = [_[_A / x]_B / z]_[_z / y]_C)
15	14	adantr 425	. . . . 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 1929	. . . 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 1927	. . 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 1350	. . . . 5 |- (A.x B e. _V -> A.xA.x B e. _V)
19		csbcog 2547	. . . . . 6 |- (B e. _V -> [_B / z]_[_z / y]_C = [_B / y]_C)
20	19	a4s 1330	. . . . 5 |- (A.x B e. _V -> [_B / z]_[_z / y]_C = [_B / y]_C)
21	18, 20	csbeq2d 2561	. . . 4 |- ((A.x B e. _V /\ A e. _V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
22	21	ancoms 484	. . 3 |- ((A e. _V /\ A.x B e. _V) -> [_A / x]_[_B / z]_[_z / y]_C = [_A / x]_[_B / y]_C)
23		csbexg 2548	. . . 4 |- ((A e. _V /\ A.x B e. _V) -> [_A / x]_B e. _V)
24		csbcog 2547	. . . 4 |- ([_A / x]_B e. _V -> [_[_A / x]_B / z]_[_z / y]_C = [_[_A / x]_B / y]_C)
25	23, 24	syl 12	. . 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 1934	. 2 |- ((A e. _V /\ A.x B e. _V) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
27		elisset 2299	. 2 |- (A e. R -> A e. _V)
28		elisset 2299	. . 3 |- (B e. S -> B e. _V)
29	28	alimi 1338	. 2 |- (A.x B e. S -> A.x B e. _V)
30	26, 27, 29	syl2an 503	1 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  sbcnestg 2583  csbco3g 2585

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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