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Mirrors > Home > ILE Home > Th. List > hbeud | GIF version |
Description: Deduction version of hbeu 2059. (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 1473 | . . 3 ⊢ Ⅎ𝑦𝜑 |
3 | hbeud.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 3 | nfi 1473 | . . . 4 ⊢ Ⅎ𝑥𝜑 |
5 | hbeud.3 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | nfd 1534 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
7 | 2, 6 | nfeud 2054 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
8 | 7 | nfrd 1531 | 1 ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 ∃!weu 2038 |
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-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 |
This theorem is referenced by: (None) |
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