Theorem csbnest1g 2957

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 2938	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
2	1	ax-gen 1378	. . 3 ⊢ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
3		csbnestgf 2954	. . 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 415	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶)
5		csbco 2917	. . 3 ⊢ ⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶
6	5	csbeq2i 2932	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶
7		csbco 2917	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶
8	4, 6, 7	3eqtr3g 2136	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1282   = wceq 1284   ∈ wcel 1433  Ⅎwnfc 2206  ⦋csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbidmg  2958