Theorem csbnest1g 2582

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
csbnest1g	|- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)

Distinct variable group:   x,A

Proof of Theorem csbnest1g

Step	Hyp	Ref	Expression
1		simpl 346	. . 3 |- ((A e. _V /\ A.x B e. S) -> A e. _V)
2		ax-17 1317	. . . . 5 |- (A e. _V -> A.x A e. _V)
3		hba1 1350	. . . . 5 |- (A.x B e. S -> A.xA.x B e. S)
4	2, 3	hban 1356	. . . 4 |- ((A e. _V /\ A.x B e. S) -> A.x(A e. _V /\ A.x B e. S))
5		csbexg 2548	. . . . . 6 |- ((A e. _V /\ A.x B e. S) -> [_A / x]_B e. _V)
6		ax-17 1317	. . . . . . . 8 |- (y e. A -> A.x y e. A)
7	6	hbcsb1g 2567	. . . . . . 7 |- (A e. _V -> (y e. [_A / x]_B -> A.x y e. [_A / x]_B))
8	2, 7	hbcsb1gd 2570	. . . . . 6 |- ((A e. _V /\ [_A / x]_B e. _V) -> (y e. [_[_A / x]_B / x]_C -> A.x y e. [_[_A / x]_B / x]_C))
9	5, 8	syldan 516	. . . . 5 |- ((A e. _V /\ A.x B e. S) -> (y e. [_[_A / x]_B / x]_C -> A.x y e. [_[_A / x]_B / x]_C))
10	9	19.21aiv 1664	. . . 4 |- ((A e. _V /\ A.x B e. S) -> A.y(y e. [_[_A / x]_B / x]_C -> A.x y e. [_[_A / x]_B / x]_C))
11	4, 10	19.21ai 1345	. . 3 |- ((A e. _V /\ A.x B e. S) -> A.xA.y(y e. [_[_A / x]_B / x]_C -> A.x y e. [_[_A / x]_B / x]_C))
12		csbeq1a 2546	. . . . . 6 |- (x = A -> B = [_A / x]_B)
13	12	csbeq1d 2544	. . . . 5 |- (x = A -> [_B / x]_C = [_[_A / x]_B / x]_C)
14	13	ax-gen 1305	. . . 4 |- A.x(x = A -> [_B / x]_C = [_[_A / x]_B / x]_C)
15	14	a1i 8	. . 3 |- ((A e. _V /\ A.x B e. S) -> A.x(x = A -> [_B / x]_C = [_[_A / x]_B / x]_C))
16		csbiegft 2573	. . 3 |- ((A e. _V /\ A.xA.y(y e. [_[_A / x]_B / x]_C -> A.x y e. [_[_A / x]_B / x]_C) /\ A.x(x = A -> [_B / x]_C = [_[_A / x]_B / x]_C)) -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)
17	1, 11, 15, 16	syl111anc 1100	. 2 |- ((A e. _V /\ A.x B e. S) -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)
18		elisset 2299	. 2 |- (A e. R -> A e. _V)
19	17, 18	sylan 497	1 |- ((A e. R /\ A.x B e. S) -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)

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

This theorem is referenced by:  csbidmg 2584  fopabcos 4806

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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