Theorem csbnestglem 2035

Description: Lemma for csbnestg 2036.

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		csbiegft 2029	. 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)
2		pm3.26 319	. 2 |- ((A e. R /\ A.x B e. S) -> A e. R)
3		ax-17 971	. . . 4 |- (A e. R -> A.x A e. R)
4		hba1 1003	. . . 4 |- (A.x B e. S -> A.xA.x B e. S)
5	3, 4	hban 1009	. . 3 |- ((A e. R /\ A.x B e. S) -> A.x(A e. R /\ A.x B e. S))
6		csbexg 2008	. . . . 5 |- ((A e. R /\ A.x B e. S) -> [_A / x]_B e. V)
7		ax-17 971	. . . . . . 7 |- (A e. R -> A.y A e. R)
8		ax-17 971	. . . . . . 7 |- (A.x B e. S -> A.yA.x B e. S)
9	7, 8	hban 1009	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> A.y(A e. R /\ A.x B e. S))
10		ax-17 971	. . . . . . . 8 |- (z e. A -> A.x z e. A)
11	10	hbcsb1g 2024	. . . . . . 7 |- (A e. R -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
12	11	adantr 389	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> (z e. [_A / x]_B -> A.x z e. [_A / x]_B))
13		ax-17 971	. . . . . . 7 |- (z e. C -> A.x z e. C)
14	13	a1i 8	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> (z e. C -> A.x z e. C))
15	5, 9, 12, 14	hbcsbgd 2028	. . . . 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))
16	6, 15	mpdan 704	. . . 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))
17	16	19.21aiv 1286	. . 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))
18	5, 17	19.21ai 998	. 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))
19		csbeq1a 2006	. . . . 5 |- (x = A -> B = [_A / x]_B)
20	19	csbeq1d 2004	. . . 4 |- (x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)
21	20	ax-gen 963	. . 3 |- A.x(x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)
22	21	a1i 8	. 2 |- ((A e. R /\ A.x B e. S) -> A.x(x = A -> [_B / y]_C = [_[_A / x]_B / y]_C))
23	1, 2, 18, 22	syl3anc 858	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:  csbnestg 2036

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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