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Mirrors > Home > ILE Home > Th. List > eumo0 | GIF version |
Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
Ref | Expression |
---|---|
eumo0.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
eumo0 | ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo0.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | euf 2011 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
3 | biimp 117 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
4 | 3 | alimi 1435 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | 4 | eximi 1580 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
6 | 2, 5 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 ∃wex 1472 ∃!weu 2006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-eu 2009 |
This theorem is referenced by: eu2 2050 eu3h 2051 |
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