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Mirrors > Home > MPE Home > Th. List > rspc8v | Structured version Visualization version GIF version |
Description: 8-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025.) |
Ref | Expression |
---|---|
rspc8v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc8v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) |
rspc8v.3 | ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) |
rspc8v.4 | ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜂)) |
rspc8v.5 | ⊢ (𝑝 = 𝐸 → (𝜂 ↔ 𝜁)) |
rspc8v.6 | ⊢ (𝑞 = 𝐹 → (𝜁 ↔ 𝜎)) |
rspc8v.7 | ⊢ (𝑟 = 𝐺 → (𝜎 ↔ 𝜌)) |
rspc8v.8 | ⊢ (𝑠 = 𝐻 → (𝜌 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc8v | ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) ∧ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌))) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc8v.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
2 | 1 | 4ralbidv 3223 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 ↔ ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜒)) |
3 | rspc8v.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
4 | 3 | 4ralbidv 3223 | . . 3 ⊢ (𝑦 = 𝐵 → (∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜒 ↔ ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜃)) |
5 | rspc8v.3 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) | |
6 | 5 | 4ralbidv 3223 | . . 3 ⊢ (𝑧 = 𝐶 → (∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜃 ↔ ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜏)) |
7 | rspc8v.4 | . . . 4 ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜂)) | |
8 | 7 | 4ralbidv 3223 | . . 3 ⊢ (𝑤 = 𝐷 → (∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜏 ↔ ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜂)) |
9 | 2, 4, 6, 8 | rspc4v 3631 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 → ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜂)) |
10 | rspc8v.5 | . . 3 ⊢ (𝑝 = 𝐸 → (𝜂 ↔ 𝜁)) | |
11 | rspc8v.6 | . . 3 ⊢ (𝑞 = 𝐹 → (𝜁 ↔ 𝜎)) | |
12 | rspc8v.7 | . . 3 ⊢ (𝑟 = 𝐺 → (𝜎 ↔ 𝜌)) | |
13 | rspc8v.8 | . . 3 ⊢ (𝑠 = 𝐻 → (𝜌 ↔ 𝜓)) | |
14 | 10, 11, 12, 13 | rspc4v 3631 | . 2 ⊢ (((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌)) → (∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜂 → 𝜓)) |
15 | 9, 14 | sylan9 509 | 1 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) ∧ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌))) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 → 𝜓)) |
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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