Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1946 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
3 | bi1 116 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
4 | 3 | alimi 1384 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | 4 | eximi 1531 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
6 | 2, 5 | sylbi 119 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∃wex 1421 ∃!weu 1941 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-eu 1944 |
This theorem is referenced by: eu2 1985 eu3h 1986 |
Copyright terms: Public domain | W3C validator |