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Theorem csbnest1g 2958
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 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)

Proof of Theorem csbnest1g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 2939 . . . 4 𝑥𝑦 / 𝑥𝐶
21ax-gen 1379 . . 3 𝑦𝑥𝑦 / 𝑥𝐶
3 csbnestgf 2955 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝑥𝑦 / 𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
42, 3mpan2 416 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
5 csbco 2918 . . 3 𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐵 / 𝑥𝐶
65csbeq2i 2933 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
7 csbco 2918 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
84, 6, 73eqtr3g 2137 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wcel 1434  wnfc 2207  csb 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by:  csbidmg  2959
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