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Theorem csbnestgw 4336
Description: Nest the composition of two substitutions. Version of csbnestg 4341 with a disjoint variable condition, which does not require ax-13 2371. (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 2904 . . 3 𝑥𝐶
21ax-gen 1803 . 2 𝑦𝑥𝐶
3 csbnestgfw 4334 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
42, 3mpan2 691 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1541   = wceq 1543  wcel 2110  wnfc 2884  csb 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410  df-sbc 3695  df-csb 3812
This theorem is referenced by:  disjxpin  30646  poimirlem24  35538  cdleme31snd  38137  cdlemeg46c  38264
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