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Theorem csbnestgw 4414
Description: Nest the composition of two substitutions. Version of csbnestg 4419 with a disjoint variable condition, which does not require ax-13 2363. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2363. (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 2895 . . 3 𝑥𝐶
21ax-gen 1789 . 2 𝑦𝑥𝐶
3 csbnestgfw 4412 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
42, 3mpan2 688 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098  wnfc 2875  csb 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468  df-sbc 3771  df-csb 3887
This theorem is referenced by:  disjxpin  32291  poimirlem24  37006  cdleme31snd  39751  cdlemeg46c  39878
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