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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 Gino Giotto, 3-Oct-2024.) |
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 | eleq1 2822 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) | |
2 | 1 | 3anbi1d 1441 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆))) |
3 | vtocl3ga.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | imbi12d 345 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) ↔ ((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜓))) |
5 | eleq1 2822 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅)) | |
6 | 5 | 3anbi2d 1442 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆))) |
7 | vtocl3ga.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
8 | 6, 7 | imbi12d 345 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜓) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜒))) |
9 | eleq1 2822 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑆 ↔ 𝐶 ∈ 𝑆)) | |
10 | 9 | 3anbi3d 1443 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆))) |
11 | vtocl3ga.3 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
12 | 10, 11 | imbi12d 345 | . . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜒) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃))) |
13 | vtocl3ga.4 | . . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | |
14 | 4, 8, 12, 13 | vtocl3g 3534 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)) |
15 | 14 | pm2.43i 52 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 |
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-v 3449 |
This theorem is referenced by: preq12bg 4815 poclOLD 5557 xpord3pred 8088 jensenlem2 26360 wrdt2ind 31863 |
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