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Theorem csbnestgfw 4380
Description: Nest the composition of two substitutions. Version of csbnestgf 4385 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
csbnestgfw ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbnestgfw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3462 . . 3 (𝐴𝑉𝐴 ∈ V)
2 df-csb 3857 . . . . . . 7 𝐵 / 𝑦𝐶 = {𝑧[𝐵 / 𝑦]𝑧𝐶}
32eqabi 2878 . . . . . 6 (𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐶)
43sbcbii 3800 . . . . 5 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶)
5 nfcr 2889 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥 𝑧𝐶)
65alimi 1814 . . . . . 6 (∀𝑦𝑥𝐶 → ∀𝑦𝑥 𝑧𝐶)
7 sbcnestgfw 4379 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦𝑥 𝑧𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
86, 7sylan2 594 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
94, 8bitrid 283 . . . 4 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
109abbidv 2802 . . 3 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
111, 10sylan 581 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
12 df-csb 3857 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶}
13 df-csb 3857 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶}
1411, 12, 133eqtr4g 2798 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wnf 1786  wcel 2107  {cab 2710  wnfc 2884  Vcvv 3444  [wsbc 3740  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:  csbnestgw  4382  csbnest1g  4390
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