Theorem csbnest1g 4386

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 3875	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
2	1	ax-gen 1797	. . 3 ⊢ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
3		csbnestgfw 4376	. . 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 692	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶)
5		csbcow 3866	. . 3 ⊢ ⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶
6	5	csbeq2i 3859	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶
7		csbcow 3866	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶
8	4, 6, 7	3eqtr3g 2795	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  ⦋csb 3851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3444  df-sbc 3743  df-csb 3852

This theorem is referenced by:  csbidm  4387