![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > hbeud | GIF version |
Description: Deduction version of hbeu 1969. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
Ref | Expression |
---|---|
hbeud.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbeud.2 | ⊢ (𝜑 → ∀𝑦𝜑) |
hbeud.3 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
hbeud | ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbeud.2 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | nfi 1396 | . . 3 ⊢ Ⅎ𝑦𝜑 |
3 | hbeud.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 3 | nfi 1396 | . . . 4 ⊢ Ⅎ𝑥𝜑 |
5 | hbeud.3 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | nfd 1461 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
7 | 2, 6 | nfeud 1964 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
8 | 7 | nfrd 1458 | 1 ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 ∃!weu 1948 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |