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Theorem csbnest1g 3025
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 3005 . . . 4 𝑥𝑦 / 𝑥𝐶
21ax-gen 1410 . . 3 𝑦𝑥𝑦 / 𝑥𝐶
3 csbnestgf 3022 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝑥𝑦 / 𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
42, 3mpan2 421 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
5 csbco 2984 . . 3 𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐵 / 𝑥𝐶
65csbeq2i 2999 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
7 csbco 2984 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
84, 6, 73eqtr3g 2173 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465  wnfc 2245  csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbidmg  3026
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