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Theorem csbnestgw 4382
Description: Nest the composition of two substitutions. Version of csbnestg 4387 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2371. (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 1798 . 2 𝑦𝑥𝐶
3 csbnestgfw 4380 . 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 1540   = wceq 1542  wcel 2107  wnfc 2884  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857
This theorem is referenced by:  disjxpin  31552  poimirlem24  36148  cdleme31snd  38895  cdlemeg46c  39022
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