Theorem csbnest1g 2034

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
csbnest1g	⊢ ((A ∈ R ⋀ ∀x B ∈ 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		csbiegft 2026	. . 3 ⊢ ((A ∈ V ⋀ ∀x∀y(y ∈ [[A / x]B / x]C → ∀x y ∈ [[A / x]B / x]C) ⋀ ∀x(x = A → [B / x]C = [[A / x]B / x]C)) → [A / x][B / x]C = [[A / x]B / x]C)
2		pm3.26 319	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → A ∈ V)
3		ax-17 970	. . . . 5 ⊢ (A ∈ V → ∀x A ∈ V)
4		hba1 1002	. . . . 5 ⊢ (∀x B ∈ S → ∀x∀x B ∈ S)
5	3, 4	hban 1008	. . . 4 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → ∀x(A ∈ V ⋀ ∀x B ∈ S))
6		csbexg 2005	. . . . . 6 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → [A / x]B ∈ V)
7		ax-17 970	. . . . . . . 8 ⊢ (y ∈ A → ∀x y ∈ A)
8	7	hbcsb1g 2021	. . . . . . 7 ⊢ (A ∈ V → (y ∈ [A / x]B → ∀x y ∈ [A / x]B))
9	3, 8	hbcsb1gd 2024	. . . . . 6 ⊢ ((A ∈ V ⋀ [A / x]B ∈ V) → (y ∈ [[A / x]B / x]C → ∀x y ∈ [[A / x]B / x]C))
10	6, 9	syldan 467	. . . . 5 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → (y ∈ [[A / x]B / x]C → ∀x y ∈ [[A / x]B / x]C))
11	10	19.21aiv 1285	. . . 4 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → ∀y(y ∈ [[A / x]B / x]C → ∀x y ∈ [[A / x]B / x]C))
12	5, 11	19.21ai 997	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → ∀x∀y(y ∈ [[A / x]B / x]C → ∀x y ∈ [[A / x]B / x]C))
13		csbeq1a 2003	. . . . . 6 ⊢ (x = A → B = [A / x]B)
14	13	csbeq1d 2001	. . . . 5 ⊢ (x = A → [B / x]C = [[A / x]B / x]C)
15	14	ax-gen 962	. . . 4 ⊢ ∀x(x = A → [B / x]C = [[A / x]B / x]C)
16	15	a1i 8	. . 3 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → ∀x(x = A → [B / x]C = [[A / x]B / x]C))
17	1, 2, 12, 16	syl3anc 857	. 2 ⊢ ((A ∈ V ⋀ ∀x B ∈ S) → [A / x][B / x]C = [[A / x]B / x]C)
18		elisset 1814	. 2 ⊢ (A ∈ R → A ∈ V)
19	17, 18	sylan 448	1 ⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [A / x][B / x]C = [[A / x]B / x]C)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  Vcvv 1808  [csb 1998

This theorem is referenced by:  csbidmg 2036  fopabcos 3828

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939  df-csb 1999

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