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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 2009 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | hba1 1520 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
3 | 2 | hbex 1616 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
4 | 1, 3 | hbxfrbi 1452 | 1 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) |
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-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-eu 2009 |
This theorem is referenced by: hbmo1 2044 eupicka 2086 exists2 2103 |
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