Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbnestgw Structured version   Visualization version   GIF version

Theorem csbnestgw 4329
 Description: Nest the composition of two substitutions. Version of csbnestg 4334 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
Assertion
Ref Expression
csbnestgw (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbnestgw
StepHypRef Expression
1 nfcv 2955 . . 3 𝑥𝐶
21ax-gen 1797 . 2 𝑦𝑥𝐶
3 csbnestgfw 4327 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
42, 3mpan2 690 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ⦋csb 3828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-csb 3829 This theorem is referenced by:  disjxpin  30361  poimirlem24  35100  cdleme31snd  37701  cdlemeg46c  37828
 Copyright terms: Public domain W3C validator