Theorem csbnest1g 3189

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 3169	. . . 4 ⊢ Ⅎx[y / x]C
2	1	ax-gen 1546	. . 3 ⊢ ∀yℲx[y / x]C
3		csbnestgf 3185	. . 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 652	. 2 ⊢ (A ∈ V → [A / x][B / y][y / x]C = [[A / x]B / y][y / x]C)
5		csbco 3146	. . 3 ⊢ [B / y][y / x]C = [B / x]C
6	5	csbeq2i 3163	. 2 ⊢ [A / x][B / y][y / x]C = [A / x][B / x]C
7		csbco 3146	. 2 ⊢ [[A / x]B / y][y / x]C = [[A / x]B / x]C
8	4, 6, 7	3eqtr3g 2408	1 ⊢ (A ∈ V → [A / x][B / x]C = [[A / x]B / x]C)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  [csb 3137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is used by:  csbnest1gOLD  3190  csbidmg  3191