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Theorem csbnestgf 3184
 Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
Assertion
Ref Expression
csbnestgf ((A V yxC) → [A / x][B / y]C = [[A / x]B / y]C)

Proof of Theorem csbnestgf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . . 3 (A VA V)
2 df-csb 3137 . . . . . . 7 [B / y]C = {z B / yz C}
32abeq2i 2460 . . . . . 6 (z [B / y]C ↔ [̣B / yz C)
43sbcbii 3101 . . . . 5 ([̣A / xz [B / y]C ↔ [̣A / x]̣[̣B / yz C)
5 nfcr 2481 . . . . . . 7 (xC → Ⅎx z C)
65alimi 1559 . . . . . 6 (yxCyx z C)
7 sbcnestgf 3183 . . . . . 6 ((A V yx z C) → ([̣A / x]̣[̣B / yz C ↔ [̣[A / x]B / yz C))
86, 7sylan2 460 . . . . 5 ((A V yxC) → ([̣A / x]̣[̣B / yz C ↔ [̣[A / x]B / yz C))
94, 8syl5bb 248 . . . 4 ((A V yxC) → ([̣A / xz [B / y]C ↔ [̣[A / x]B / yz C))
109abbidv 2467 . . 3 ((A V yxC) → {z A / xz [B / y]C} = {z [A / x]B / yz C})
111, 10sylan 457 . 2 ((A V yxC) → {z A / xz [B / y]C} = {z [A / x]B / yz C})
12 df-csb 3137 . 2 [A / x][B / y]C = {z A / xz [B / y]C}
13 df-csb 3137 . 2 [[A / x]B / y]C = {z [A / x]B / yz C}
1411, 12, 133eqtr4g 2410 1 ((A V yxC) → [A / x][B / y]C = [[A / x]B / y]C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  Vcvv 2859  [̣wsbc 3046  [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:  csbnestg  3186  csbnest1g  3188
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