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Mirrors > Home > ILE Home > Th. List > spc3egv | GIF version |
Description: Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) |
Ref | Expression |
---|---|
spc3egv.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spc3egv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2633 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | elisset 2633 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
3 | elisset 2633 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → ∃𝑧 𝑧 = 𝐶) | |
4 | 1, 2, 3 | 3anim123i 1128 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) |
5 | eeeanv 1856 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) | |
6 | 4, 5 | sylibr 132 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶)) |
7 | spc3egv.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
8 | 7 | biimprcd 158 | . . . 4 ⊢ (𝜓 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜑)) |
9 | 8 | eximdv 1808 | . . 3 ⊢ (𝜓 → (∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧𝜑)) |
10 | 9 | 2eximdv 1810 | . 2 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧𝜑)) |
11 | 6, 10 | syl5com 29 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 924 = wceq 1289 ∃wex 1426 ∈ wcel 1438 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-v 2621 |
This theorem is referenced by: (None) |
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