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| Mirrors > Home > MPE Home > Th. List > vtocl3ga | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by GG, 3-Oct-2024.) (Proof shortened by Wolf Lammen, 31-May-2025.) | 
| Ref | Expression | 
|---|---|
| vtocl3ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| vtocl3ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | 
| vtocl3ga.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | 
| vtocl3ga.4 | ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | 
| Ref | Expression | 
|---|---|
| vtocl3ga | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vtocl3ga.3 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
| 2 | 1 | imbi2d 340 | . . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜃))) | 
| 3 | vtocl3ga.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | imbi2d 340 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ 𝑆 → 𝜑) ↔ (𝑧 ∈ 𝑆 → 𝜓))) | 
| 5 | vtocl3ga.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 6 | 5 | imbi2d 340 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ 𝑆 → 𝜓) ↔ (𝑧 ∈ 𝑆 → 𝜒))) | 
| 7 | vtocl3ga.4 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | |
| 8 | 7 | 3expia 1121 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅) → (𝑧 ∈ 𝑆 → 𝜑)) | 
| 9 | 4, 6, 8 | vtocl2ga 3577 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → (𝑧 ∈ 𝑆 → 𝜒)) | 
| 10 | 9 | com12 32 | . . . 4 ⊢ (𝑧 ∈ 𝑆 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜒)) | 
| 11 | 2, 10 | vtoclga 3576 | . . 3 ⊢ (𝐶 ∈ 𝑆 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜃)) | 
| 12 | 11 | impcom 407 | . 2 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) ∧ 𝐶 ∈ 𝑆) → 𝜃) | 
| 13 | 12 | 3impa 1109 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: preq12bg 4852 xpord3pred 8178 jensenlem2 27032 wrdt2ind 32939 | 
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