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Mirrors > Home > MPE Home > Th. List > spc3gv | Structured version Visualization version GIF version |
Description: 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 |
---|---|
spc3gv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spc3egv.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 320 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | spc3egv 3606 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (¬ 𝜓 → ∃𝑥∃𝑦∃𝑧 ¬ 𝜑)) |
4 | exnal 1827 | . . . . . . 7 ⊢ (∃𝑧 ¬ 𝜑 ↔ ¬ ∀𝑧𝜑) | |
5 | 4 | exbii 1848 | . . . . . 6 ⊢ (∃𝑦∃𝑧 ¬ 𝜑 ↔ ∃𝑦 ¬ ∀𝑧𝜑) |
6 | exnal 1827 | . . . . . 6 ⊢ (∃𝑦 ¬ ∀𝑧𝜑 ↔ ¬ ∀𝑦∀𝑧𝜑) | |
7 | 5, 6 | bitri 277 | . . . . 5 ⊢ (∃𝑦∃𝑧 ¬ 𝜑 ↔ ¬ ∀𝑦∀𝑧𝜑) |
8 | 7 | exbii 1848 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑧 ¬ 𝜑 ↔ ∃𝑥 ¬ ∀𝑦∀𝑧𝜑) |
9 | exnal 1827 | . . . 4 ⊢ (∃𝑥 ¬ ∀𝑦∀𝑧𝜑 ↔ ¬ ∀𝑥∀𝑦∀𝑧𝜑) | |
10 | 8, 9 | bitr2i 278 | . . 3 ⊢ (¬ ∀𝑥∀𝑦∀𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧 ¬ 𝜑) |
11 | 3, 10 | syl6ibr 254 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (¬ 𝜓 → ¬ ∀𝑥∀𝑦∀𝑧𝜑)) |
12 | 11 | con4d 115 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-v 3498 |
This theorem is referenced by: funopg 6391 pslem 17818 dirtr 17848 mclsax 32818 fununiq 33014 |
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