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Theorem sbcnestgfw 4329
 Description: Nest the composition of two substitutions. Version of sbcnestgf 4334 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 11-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
Assertion
Ref Expression
sbcnestgfw ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcnestgfw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3724 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2 csbeq1 3833 . . . . . 6 (𝑧 = 𝐴𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
32sbceq1d 3727 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
41, 3bibi12d 349 . . . 4 (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
54imbi2d 344 . . 3 (𝑧 = 𝐴 → ((∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑)) ↔ (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))))
6 vex 3445 . . . . 5 𝑧 ∈ V
76a1i 11 . . . 4 (∀𝑦𝑥𝜑𝑧 ∈ V)
8 csbeq1a 3844 . . . . . 6 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
98sbceq1d 3727 . . . . 5 (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
109adantl 485 . . . 4 ((∀𝑦𝑥𝜑𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
11 nfnf1 2155 . . . . 5 𝑥𝑥𝜑
1211nfal 2331 . . . 4 𝑥𝑦𝑥𝜑
13 nfa1 2152 . . . . 5 𝑦𝑦𝑥𝜑
14 nfcsb1v 3854 . . . . . 6 𝑥𝑧 / 𝑥𝐵
1514a1i 11 . . . . 5 (∀𝑦𝑥𝜑𝑥𝑧 / 𝑥𝐵)
16 sp 2180 . . . . 5 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝜑)
1713, 15, 16nfsbcdw 3743 . . . 4 (∀𝑦𝑥𝜑 → Ⅎ𝑥[𝑧 / 𝑥𝐵 / 𝑦]𝜑)
187, 10, 12, 17sbciedf 3763 . . 3 (∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
195, 18vtoclg 3516 . 2 (𝐴𝑉 → (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
2019imp 410 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936  Vcvv 3442  [wsbc 3722  ⦋csb 3830 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 3444  df-sbc 3723  df-csb 3831 This theorem is referenced by:  csbnestgfw  4330  sbcnestgw  4331
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