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Mirrors > Home > ILE Home > Th. List > moaneu | GIF version |
Description: Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) |
Ref | Expression |
---|---|
moaneu | ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 1973 | . . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | nfeu1 1952 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
3 | 2 | moanim 2015 | . . 3 ⊢ (∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑)) |
4 | 1, 3 | mpbir 144 | . 2 ⊢ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) |
5 | ancom 262 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑 ∧ 𝜑)) | |
6 | 5 | mobii 1978 | . 2 ⊢ (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑)) |
7 | 4, 6 | mpbir 144 | 1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∃!weu 1941 ∃*wmo 1942 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 |
This theorem is referenced by: (None) |
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