Theorem csbnestg 3187

Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Assertion

Ref	Expression
csbnestg	⊢ (A ∈ V → [A / x][B / y]C = [[A / x]B / y]C)

Distinct variable group:   x,C

Allowed substitution hints:   A(x,y)   B(x,y)   C(y)   V(x,y)

Proof of Theorem csbnestg

Step	Hyp	Ref	Expression
1		nfcv 2490	. . 3 ⊢ ℲxC
2	1	ax-gen 1546	. 2 ⊢ ∀yℲxC
3		csbnestgf 3185	. 2 ⊢ ((A ∈ V ∧ ∀yℲxC) → [A / x][B / y]C = [[A / x]B / y]C)
4	2, 3	mpan2 652	1 ⊢ (A ∈ V → [A / x][B / y]C = [[A / x]B / y]C)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  [csb 3137

This theorem was proved from 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 theorem 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 referenced by:  csbnestgOLD  3188  csbco3g  3194