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Mirrors > Home > ILE Home > Th. List > mo2r | GIF version |
Description: A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
Ref | Expression |
---|---|
mo2r.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo2r | ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo2r.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1453 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | eu3h 1988 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | 3 | simplbi2com 1374 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | df-mo 1947 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
6 | 4, 5 | sylibr 132 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1283 Ⅎwnf 1390 ∃wex 1422 ∃!weu 1943 ∃*wmo 1944 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 |
This theorem is referenced by: mo2icl 2782 rmo2ilem 2914 dffun5r 4981 frecuzrdgtcl 9708 frecuzrdgfunlem 9715 |
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