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Mirrors > Home > ILE Home > Th. List > hbeu1 | GIF version |
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
hbeu1 | ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 1903 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | hba1 1433 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
3 | 2 | hbex 1527 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
4 | 1, 3 | hbxfrbi 1361 | 1 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 ∃wex 1381 ∃!weu 1900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-eu 1903 |
This theorem is referenced by: hbmo1 1938 eupicka 1980 exists2 1997 |
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