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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 2642 for a condition under which the naive definition holds and 2exeu 2638 for a one-way implication. See 2eu5 2647 and 2eu8 2650 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 2559 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
2 | df-eu 2559 | . . . 4 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
3 | excom 2152 | . . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
4 | 2, 3 | bianbi 626 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) |
5 | 1, 4 | anbi12i 627 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))) |
6 | anandi 675 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))) | |
7 | 2mo2 2639 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | |
8 | 7 | anbi2i 622 | . 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 205 ∧ wa 395 ∀wal 1532 ∃wex 1774 ∃*wmo 2528 ∃!weu 2558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-11 2147 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-mo 2530 df-eu 2559 |
This theorem is referenced by: 2eu5 2647 2eu6 2648 |
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