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Mirrors > Home > ILE Home > Th. List > hbsbd | GIF version |
Description: Deduction version of hbsb 1823. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Ref | Expression |
---|---|
hbsbd.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbsbd.2 | ⊢ (𝜑 → ∀𝑧𝜑) |
hbsbd.3 | ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) |
Ref | Expression |
---|---|
hbsbd | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsbd.2 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | nfi 1351 | . . 3 ⊢ Ⅎ𝑧𝜑 |
3 | hbsbd.3 | . . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) | |
4 | 1, 3 | nfdh 1417 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) |
5 | 2, 4 | nfim1 1463 | . . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓) |
6 | 5 | nfsb 1822 | . . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) |
7 | hbsbd.1 | . . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑) | |
8 | 7 | sbrim 1830 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
9 | 8 | nfbii 1362 | . . . 4 ⊢ (Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓)) |
10 | 6, 9 | mpbi 133 | . . 3 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓) |
11 | 2, 10 | nfrimi 1418 | . 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
12 | 11 | nfrd 1413 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: (None) |
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