Theorem csbnest1g 4431

Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
csbnest1g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)

Proof of Theorem csbnest1g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcsb1v 3914	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
2	1	ax-gen 1789	. . 3 ⊢ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
3		csbnestgfw 4421	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶)
4	2, 3	mpan2 689	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶)
5		csbcow 3904	. . 3 ⊢ ⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶
6	5	csbeq2i 3897	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶
7		csbcow 3904	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶
8	4, 6, 7	3eqtr3g 2788	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2875  ⦋csb 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-v 3463  df-sbc 3774  df-csb 3890

This theorem is referenced by:  csbidm  4432