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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 2030 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
2 | nfeud.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeud.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | nfexd 1761 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
5 | 2, 3 | nfeud 2042 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
6 | 4, 5 | nfimd 1585 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | 1, 6 | nfxfrd 1475 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1460 ∃wex 1492 ∃!weu 2026 ∃*wmo 2027 |
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-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 |
This theorem is referenced by: nfmo 2046 |
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