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Theorem csbnest1g 3188
 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.
StepHypRef Expression
1 nfcsb1v 3168 . . . 4 x[y / x]C
21ax-gen 1546 . . 3 yx[y / x]C
3 csbnestgf 3184 . . 3 ((A V yx[y / x]C) → [A / x][B / y][y / x]C = [[A / x]B / y][y / x]C)
42, 3mpan2 652 . 2 (A V[A / x][B / y][y / x]C = [[A / x]B / y][y / x]C)
5 csbco 3145 . . 3 [B / y][y / x]C = [B / x]C
65csbeq2i 3162 . 2 [A / x][B / y][y / x]C = [A / x][B / x]C
7 csbco 3145 . 2 [[A / x]B / y][y / x]C = [[A / x]B / x]C
84, 6, 73eqtr3g 2408 1 (A V[A / x][B / x]C = [[A / x]B / x]C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbnest1gOLD  3189  csbidmg  3190
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