Theorem csbnestg 2026

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
csbnestg	⊢ ((A ∈ R ⋀ ∀x B ∈ 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 1997	. . . . 5 ⊢ (A ∈ V → [A / w][w / x][B / z][z / y]C = [A / x][B / z][z / y]C)
2	1	adantr 389	. . . 4 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / w][w / x][B / z][z / y]C = [A / x][B / z][z / y]C)
3		visset 1804	. . . . . . . 8 ⊢ w ∈ V
4		csbnestglem 2025	. . . . . . . 8 ⊢ ((w ∈ V ⋀ ∀x B ∈ V) → [w / x][B / z][z / y]C = [[w / x]B / z][z / y]C)
5	3, 4	mpan 693	. . . . . . 7 ⊢ (∀x B ∈ V → [w / x][B / z][z / y]C = [[w / x]B / z][z / y]C)
6	5	csbeq2dv 2009	. . . . . 6 ⊢ ((∀x B ∈ V ⋀ A ∈ V) → [A / w][w / x][B / z][z / y]C = [A / w][[w / x]B / z][z / y]C)
7	6	ancoms 436	. . . . 5 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / w][w / x][B / z][z / y]C = [A / w][[w / x]B / z][z / y]C)
8		csbnestglem 2025	. . . . . 6 ⊢ ((A ∈ V ⋀ ∀w[w / x]B ∈ V) → [A / w][[w / x]B / z][z / y]C = [[A / w][w / x]B / z][z / y]C)
9		csbexg 1998	. . . . . . . 8 ⊢ ((w ∈ V ⋀ ∀x B ∈ V) → [w / x]B ∈ V)
10	3, 9	mpan 693	. . . . . . 7 ⊢ (∀x B ∈ V → [w / x]B ∈ V)
11	10	19.21aiv 1281	. . . . . 6 ⊢ (∀x B ∈ V → ∀w[w / x]B ∈ V)
12	8, 11	sylan2 451	. . . . 5 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / w][[w / x]B / z][z / y]C = [[A / w][w / x]B / z][z / y]C)
13		csbcog 1997	. . . . . . 7 ⊢ (A ∈ V → [A / w][w / x]B = [A / x]B)
14	13	csbeq1d 1994	. . . . . 6 ⊢ (A ∈ V → [[A / w][w / x]B / z][z / y]C = [[A / x]B / z][z / y]C)
15	14	adantr 389	. . . . 5 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [[A / w][w / x]B / z][z / y]C = [[A / x]B / z][z / y]C)
16	7, 12, 15	3eqtrd 1503	. . . 4 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / w][w / x][B / z][z / y]C = [[A / x]B / z][z / y]C)
17	2, 16	eqtr3d 1501	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / x][B / z][z / y]C = [[A / x]B / z][z / y]C)
18		hba1 1000	. . . . 5 ⊢ (∀x B ∈ V → ∀x∀x B ∈ V)
19		csbcog 1997	. . . . . 6 ⊢ (B ∈ V → [B / z][z / y]C = [B / y]C)
20	19	a4s 981	. . . . 5 ⊢ (∀x B ∈ V → [B / z][z / y]C = [B / y]C)
21	18, 20	csbeq2d 2008	. . . 4 ⊢ ((∀x B ∈ V ⋀ A ∈ V) → [A / x][B / z][z / y]C = [A / x][B / y]C)
22	21	ancoms 436	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / x][B / z][z / y]C = [A / x][B / y]C)
23		csbexg 1998	. . . 4 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / x]B ∈ V)
24		csbcog 1997	. . . 4 ⊢ ([A / x]B ∈ V → [[A / x]B / z][z / y]C = [[A / x]B / y]C)
25	23, 24	syl 10	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [[A / x]B / z][z / y]C = [[A / x]B / y]C)
26	17, 22, 25	3eqtr3d 1507	. 2 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / x][B / y]C = [[A / x]B / y]C)
27		elisset 1808	. 2 ⊢ (A ∈ R → A ∈ V)
28		elisset 1808	. . 3 ⊢ (B ∈ S → B ∈ V)
29	28	19.20i 989	. 2 ⊢ (∀x B ∈ S → ∀x B ∈ V)
30	26, 27, 29	syl2an 454	1 ⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [A / x][B / y]C = [[A / x]B / y]C)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 951   = wceq 953   ∈ wcel 955  Vcvv 1802  [csb 1991

This theorem is referenced by:  sbcnestg 2028  csbco3g 2030

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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