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Theorem rspc8v 3633
Description: 8-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025.)
Hypotheses
Ref Expression
rspc8v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc8v.2 (𝑦 = 𝐵 → (𝜒𝜃))
rspc8v.3 (𝑧 = 𝐶 → (𝜃𝜏))
rspc8v.4 (𝑤 = 𝐷 → (𝜏𝜂))
rspc8v.5 (𝑝 = 𝐸 → (𝜂𝜁))
rspc8v.6 (𝑞 = 𝐹 → (𝜁𝜎))
rspc8v.7 (𝑟 = 𝐺 → (𝜎𝜌))
rspc8v.8 (𝑠 = 𝐻 → (𝜌𝜓))
Assertion
Ref Expression
rspc8v ((((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑇𝐷𝑈)) ∧ ((𝐸𝑉𝐹𝑊) ∧ (𝐺𝑋𝐻𝑌))) → (∀𝑥𝑅𝑦𝑆𝑧𝑇𝑤𝑈𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜑𝜓))
Distinct variable groups:   𝐴,𝑝,𝑞,𝑟,𝑠,𝑤,𝑥,𝑦,𝑧   𝐵,𝑝,𝑞,𝑟,𝑠,𝑤,𝑦,𝑧   𝐶,𝑝,𝑞,𝑟,𝑠,𝑤,𝑧   𝐷,𝑝,𝑞,𝑟,𝑠,𝑤   𝐸,𝑝,𝑞,𝑟,𝑠   𝐹,𝑞,𝑟,𝑠   𝐺,𝑟,𝑠   𝐻,𝑠   𝑥,𝑅   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦,𝑧   𝑤,𝑈,𝑥,𝑦,𝑧   𝑉,𝑝,𝑤,𝑥,𝑦,𝑧   𝑊,𝑝,𝑞,𝑤,𝑥,𝑦,𝑧   𝑋,𝑝,𝑞,𝑟,𝑤,𝑥,𝑦,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,𝑤,𝑥,𝑦,𝑧   𝜒,𝑥   𝜃,𝑦   𝜏,𝑧   𝜂,𝑤   𝜁,𝑝   𝜎,𝑞   𝜌,𝑟   𝜓,𝑠
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑠,𝑟,𝑞,𝑝)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑟,𝑞,𝑝)   𝜒(𝑦,𝑧,𝑤,𝑠,𝑟,𝑞,𝑝)   𝜃(𝑥,𝑧,𝑤,𝑠,𝑟,𝑞,𝑝)   𝜏(𝑥,𝑦,𝑤,𝑠,𝑟,𝑞,𝑝)   𝜂(𝑥,𝑦,𝑧,𝑠,𝑟,𝑞,𝑝)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑠,𝑟,𝑞)   𝜎(𝑥,𝑦,𝑧,𝑤,𝑠,𝑟,𝑝)   𝜌(𝑥,𝑦,𝑧,𝑤,𝑠,𝑞,𝑝)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑧)   𝑅(𝑦,𝑧,𝑤,𝑠,𝑟,𝑞,𝑝)   𝑆(𝑧,𝑤,𝑠,𝑟,𝑞,𝑝)   𝑇(𝑤,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑠,𝑟,𝑞,𝑝)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑝)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑞,𝑝)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑟,𝑞,𝑝)   𝑉(𝑠,𝑟,𝑞)   𝑊(𝑠,𝑟)   𝑋(𝑠)

Proof of Theorem rspc8v
StepHypRef Expression
1 rspc8v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
214ralbidv 3223 . . 3 (𝑥 = 𝐴 → (∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜑 ↔ ∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜒))
3 rspc8v.2 . . . 4 (𝑦 = 𝐵 → (𝜒𝜃))
434ralbidv 3223 . . 3 (𝑦 = 𝐵 → (∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜒 ↔ ∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜃))
5 rspc8v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜏))
654ralbidv 3223 . . 3 (𝑧 = 𝐶 → (∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜃 ↔ ∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜏))
7 rspc8v.4 . . . 4 (𝑤 = 𝐷 → (𝜏𝜂))
874ralbidv 3223 . . 3 (𝑤 = 𝐷 → (∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜏 ↔ ∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜂))
92, 4, 6, 8rspc4v 3631 . 2 (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑇𝐷𝑈)) → (∀𝑥𝑅𝑦𝑆𝑧𝑇𝑤𝑈𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜑 → ∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜂))
10 rspc8v.5 . . 3 (𝑝 = 𝐸 → (𝜂𝜁))
11 rspc8v.6 . . 3 (𝑞 = 𝐹 → (𝜁𝜎))
12 rspc8v.7 . . 3 (𝑟 = 𝐺 → (𝜎𝜌))
13 rspc8v.8 . . 3 (𝑠 = 𝐻 → (𝜌𝜓))
1410, 11, 12, 13rspc4v 3631 . 2 (((𝐸𝑉𝐹𝑊) ∧ (𝐺𝑋𝐻𝑌)) → (∀𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜂𝜓))
159, 14sylan9 509 1 ((((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑇𝐷𝑈)) ∧ ((𝐸𝑉𝐹𝑊) ∧ (𝐺𝑋𝐻𝑌))) → (∀𝑥𝑅𝑦𝑆𝑧𝑇𝑤𝑈𝑝𝑉𝑞𝑊𝑟𝑋𝑠𝑌 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063
This theorem is referenced by: (None)
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