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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eumo0 | Structured version Visualization version GIF version |
Description: Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2487 and mo2 2467. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.) |
Ref | Expression |
---|---|
bj-eumo0.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
bj-eumo0 | ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-eumo0.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | euf 2466 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
3 | biimp 204 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
4 | 3 | alimi 1730 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | 4 | eximi 1752 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
6 | 2, 5 | sylbi 206 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 ∃!weu 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 |
This theorem is referenced by: (None) |
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