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| Mirrors > Home > MPE Home > Th. List > 2eu4 | Structured version Visualization version GIF version | ||
| Description: This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2650 for a condition under which the naive definition holds and 2exeu 2645 for a one-way implication. See 2eu5 2655 and 2eu8 2658 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.) | 
| Ref | Expression | 
|---|---|
| 2eu4 | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-eu 2568 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
| 2 | df-eu 2568 | . . . 4 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
| 3 | excom 2161 | . . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
| 4 | 2, 3 | bianbi 627 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | 
| 5 | 1, 4 | anbi12i 628 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))) | 
| 6 | anandi 676 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))) | |
| 7 | 2mo2 2646 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | |
| 8 | 7 | anbi2i 623 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | 
| 9 | 5, 6, 8 | 3bitr2i 299 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 ∃*wmo 2537 ∃!weu 2567 | 
| 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-11 2156 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: 2eu5 2655 2eu6 2656 | 
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