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Mirrors > Home > ILE Home > Th. List > euex | GIF version |
Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
euex | ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1513 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | eu1 2038 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) |
3 | exsimpl 1604 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) → ∃𝑥𝜑) | |
4 | 2, 3 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 ∃wex 1479 [wsb 1749 ∃!weu 2013 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-eu 2016 |
This theorem is referenced by: eu2 2057 eu3h 2058 eu5 2060 exmoeudc 2076 eupickbi 2095 2eu2ex 2102 euxfrdc 2907 repizf 4092 eusvnf 4425 eusvnfb 4426 tz6.12c 5510 ndmfvg 5511 nfvres 5513 0fv 5515 eusvobj2 5822 fnoprabg 5934 txcn 12822 |
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