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Mirrors > Home > ILE Home > Th. List > exmoeu2 | GIF version |
Description: Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
Ref | Expression |
---|---|
exmoeu2 | ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu5 2085 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
2 | 1 | baibr 921 | 1 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∃wex 1503 ∃!weu 2038 ∃*wmo 2039 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 |
This theorem is referenced by: n0mmoeu 3454 fneu 5342 |
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