Theorem csbnestglem 2025

Description: Lemma for csbnestg 2026.

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 2019	. 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-17 968	. . . 4 |- (A e. R -> A.x A e. R)
4		hba1 1000	. . . 4 |- (A.x B e. S -> A.xA.x B e. S)
5	3, 4	hban 1006	. . 3 |- ((A e. R /\ A.x B e. S) -> A.x(A e. R /\ A.x B e. S))
6		csbexg 1998	. . . . 5 |- ((A e. R /\ A.x B e. S) -> [_A / x]_B e. V)
7		ax-17 968	. . . . . . 7 |- (A e. R -> A.y A e. R)
8		ax-17 968	. . . . . . 7 |- (A.x B e. S -> A.yA.x B e. S)
9	7, 8	hban 1006	. . . . . 6 |- ((A e. R /\ A.x B e. S) -> A.y(A e. R /\ A.x B e. S))
10		ax-17 968	. . . . . . . 8 |- (z e. A -> A.x z e. A)
11	10	hbcsb1g 2014	. . . . . . 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-17 968	. . . . . . 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 2018	. . . . 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 702	. . . 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 1281	. . 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 995	. 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 1996	. . . . 5 |- (x = A -> B = [_A / x]_B)
20	19	csbeq1d 1994	. . . 4 |- (x = A -> [_B / y]_C = [_[_A / x]_B / y]_C)
21	20	ax-gen 960	. . 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 856	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 951   = wceq 953   e. wcel 955  Vcvv 1802  [_csb 1991

This theorem is referenced by:  csbnestg 2026

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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