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Theorem sbcnestgw 4420
Description: Nest the composition of two substitutions. Version of sbcnestg 4425 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.)
Assertion
Ref Expression
sbcnestgw (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcnestgw
StepHypRef Expression
1 nfv 1916 . . 3 𝑥𝜑
21ax-gen 1796 . 2 𝑦𝑥𝜑
3 sbcnestgfw 4418 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
42, 3mpan2 688 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538  wnf 1784  wcel 2105  [wsbc 3777  csb 3893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894
This theorem is referenced by:  sbcco3gw  4422  sbcop  5489
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