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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 1980 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | hba1 1505 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
3 | 2 | hbex 1600 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
4 | 1, 3 | hbxfrbi 1433 | 1 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 ∃wex 1453 ∃!weu 1977 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 df-eu 1980 |
This theorem is referenced by: hbmo1 2015 eupicka 2057 exists2 2074 |
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