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Mirrors > Home > MPE Home > Th. List > rspc3dv | Structured version Visualization version GIF version |
Description: 3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025.) |
Ref | Expression |
---|---|
rspc3dv.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
rspc3dv.2 | ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜏)) |
rspc3dv.3 | ⊢ (𝑧 = 𝐶 → (𝜏 ↔ 𝜒)) |
rspc3dv.4 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓) |
rspc3dv.5 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
rspc3dv.6 | ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
rspc3dv.7 | ⊢ (𝜑 → 𝐶 ∈ 𝐹) |
Ref | Expression |
---|---|
rspc3dv | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc3dv.5 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | rspc3dv.6 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
3 | rspc3dv.7 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐹) | |
4 | 1, 2, 3 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) |
5 | rspc3dv.4 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓) | |
6 | rspc3dv.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) | |
7 | rspc3dv.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜏)) | |
8 | rspc3dv.3 | . . 3 ⊢ (𝑧 = 𝐶 → (𝜏 ↔ 𝜒)) | |
9 | 6, 7, 8 | rspc3v 3638 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓 → 𝜒)) |
10 | 4, 5, 9 | sylc 65 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 |
This theorem is referenced by: domnlcanb 20737 domnrcanb 20739 mulsasslem3 28206 domnlcanOLD 33264 |
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