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Theorem csbnestgfw 4327
 Description: Nest the composition of two substitutions. Version of csbnestgf 4332 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 23-Nov-2005.) (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 3459 . . 3 (𝐴𝑉𝐴 ∈ V)
2 df-csb 3829 . . . . . . 7 𝐵 / 𝑦𝐶 = {𝑧[𝐵 / 𝑦]𝑧𝐶}
32abeq2i 2925 . . . . . 6 (𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐶)
43sbcbii 3776 . . . . 5 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶)
5 nfcr 2941 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥 𝑧𝐶)
65alimi 1813 . . . . . 6 (∀𝑦𝑥𝐶 → ∀𝑦𝑥 𝑧𝐶)
7 sbcnestgfw 4326 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦𝑥 𝑧𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
86, 7sylan2 595 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
94, 8syl5bb 286 . . . 4 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
109abbidv 2862 . . 3 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
111, 10sylan 583 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
12 df-csb 3829 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶}
13 df-csb 3829 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶}
1411, 12, 133eqtr4g 2858 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  Vcvv 3441  [wsbc 3720  ⦋csb 3828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-csb 3829 This theorem is referenced by:  csbnestgw  4329  csbnest1g  4337
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