Theorem csbnestglem 2580

Description: Lemma for csbnestg 2581.

Assertion

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

Distinct variable groups:   x,y,A   y,B   x,C   x,R,y   y,S

Proof of Theorem csbnestglem

Step	Hyp	Ref	Expression
1		simpl 346	. 2 |- ((A e. R /\ A.x B e. S) -> A e. R)
2		ax-17 1317	. . . 4 |- (A e. R -> A.x A e. R)
3		hba1 1350	. . . 4 |- (A.x B e. S -> A.xA.x B e. S)
4	2, 3	hban 1356	. . 3 |- ((A e. R /\ A.x B e. S) -> A.x(A e. R /\ A.x B e. S))
5		csbexg 2548	. . . . 5 |- ((A e. R /\ A.x B e. S) -> [_A / x]_B e. _V)
6		ax-17 1317	. . . . . . 7 |- (A e. R -> A.y A e. R)
7		ax-17 1317	. . . . . . 7 |- (A.x B e. S -> A.yA.x B e. S)
8	6, 7	hban 1356	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> A.y(A e. R /\ A.x B e. S))
9		ax-17 1317	. . . . . . . 8 |- (z e. A -> A.x z e. A)
10	9	hbcsb1g 2567	. . . . . . 7 |- (A e. R -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
11	10	adantr 425	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
12		ax-17 1317	. . . . . . 7 |- (z e. C -> A.x z e. C)
13	12	a1i 8	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> (z e. C -> A.x z e. C))
14	4, 8, 11, 13	hbcsbgd 2571	. . . . 5 |- (((A e. R /\ A.x B e. S) /\ [_A / x]_B e. _V) -> (z e. [_[_A / x]_B / y]_C -> A.x z e. [_[_A / x]_B / y]_C))
15	5, 14	mpdan 768	. . . 4 |- ((A e. R /\ A.x B e. S) -> (z e. [_[_A / x]_B / y]_C -> A.x z e. [_[_A / x]_B / y]_C))
16	15	19.21aiv 1664	. . 3 |- ((A e. R /\ A.x B e. S) -> A.z(z e. [_[_A / x]_B / y]_C -> A.x z e. [_[_A / x]_B / y]_C))
17	4, 16	19.21ai 1345	. 2 |- ((A e. R /\ A.x B e. S) -> A.xA.z(z e. [_[_A / x]_B / y]_C -> A.x z e. [_[_A / x]_B / y]_C))
18		csbeq1a 2546	. . . . 5 |- (x = A -> B = [_A / x]_B)
19	18	csbeq1d 2544	. . . 4 |- (x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)
20	19	ax-gen 1305	. . 3 |- A.x(x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)
21	20	a1i 8	. 2 |- ((A e. R /\ A.x B e. S) -> A.x(x = A -> [_B / y]_C = [_[_A / x]_B / y]_C))
22		csbiegft 2573	. 2 |- ((A e. R /\ A.xA.z(z e. [_[_A / x]_B / y]_C -> A.x z e. [_[_A / x]_B / y]_C) /\ A.x(x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)
23	1, 17, 21, 22	syl111anc 1100	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:  csbnestg 2581

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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