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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 1526 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | eu1 2051 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) |
3 | exsimpl 1617 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) → ∃𝑥𝜑) | |
4 | 2, 3 | sylbi 121 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 ∃wex 1492 [wsb 1762 ∃!weu 2026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 |
This theorem is referenced by: eu2 2070 eu3h 2071 eu5 2073 exmoeudc 2089 eupickbi 2108 2eu2ex 2115 euxfrdc 2923 repizf 4116 eusvnf 4450 eusvnfb 4451 tz6.12c 5541 ndmfvg 5542 nfvres 5544 0fv 5546 eusvobj2 5855 fnoprabg 5970 0g0 12684 txcn 13435 |
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