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Mirrors > Home > ILE Home > Th. List > nfmod | GIF version |
Description: Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
nfeud.1 | ⊢ Ⅎ𝑦𝜑 |
nfeud.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfmod | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 1945 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
2 | nfeud.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeud.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | nfexd 1684 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
5 | 2, 3 | nfeud 1957 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
6 | 4, 5 | nfimd 1517 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | 1, 6 | nfxfrd 1404 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1389 ∃wex 1421 ∃!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: nfmo 1961 |
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