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Mirrors > Home > MPE Home > Th. List > spc3egv | Structured version Visualization version GIF version |
Description: Existential specialization with three 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 3417 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | elisset 3417 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
3 | elisset 3417 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → ∃𝑧 𝑧 = 𝐶) | |
4 | 1, 2, 3 | 3anim123i 1151 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) |
5 | eeeanv 2319 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) | |
6 | 4, 5 | sylibr 226 | . 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 242 | . . . 4 ⊢ (𝜓 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜑)) |
9 | 8 | eximdv 1960 | . . 3 ⊢ (𝜓 → (∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧𝜑)) |
10 | 9 | 2eximdv 1962 | . 2 ⊢ (𝜓 → (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧𝜑)) |
11 | 6, 10 | syl5com 31 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-v 3400 |
This theorem is referenced by: spc3gv 3500 dihjatcclem4 37580 |
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