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Theorem csbnest1g 2972
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 2952 . . . 4 𝑥𝑦 / 𝑥𝐶
21ax-gen 1381 . . 3 𝑦𝑥𝑦 / 𝑥𝐶
3 csbnestgf 2969 . . 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 2931 . . 3 𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐵 / 𝑥𝐶
65csbeq2i 2946 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
7 csbco 2931 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
84, 6, 73eqtr3g 2140 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285   = wceq 1287  wcel 1436  wnfc 2212  csb 2922
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923
This theorem is referenced by:  csbidmg  2973
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