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Theorem csbnest1g 3127
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 3105 . . . 4 𝑥𝑦 / 𝑥𝐶
21ax-gen 1460 . . 3 𝑦𝑥𝑦 / 𝑥𝐶
3 csbnestgf 3124 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝑥𝑦 / 𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
42, 3mpan2 425 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
5 csbco 3082 . . 3 𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐵 / 𝑥𝐶
65csbeq2i 3099 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
7 csbco 3082 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
84, 6, 73eqtr3g 2245 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wcel 2160  wnfc 2319  csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  csbidmg  3128
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