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Mirrors > Home > ILE Home > Th. List > hbsbd | GIF version |
Description: Deduction version of hbsb 1929. (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 1442 | . . 3 ⊢ Ⅎ𝑧𝜑 |
3 | hbsbd.3 | . . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) | |
4 | 1, 3 | nfdh 1504 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) |
5 | 2, 4 | nfim1 1551 | . . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓) |
6 | 5 | nfsb 1926 | . . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) |
7 | hbsbd.1 | . . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑) | |
8 | 7 | sbrim 1936 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
9 | 8 | nfbii 1453 | . . . 4 ⊢ (Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓)) |
10 | 6, 9 | mpbi 144 | . . 3 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓) |
11 | 2, 10 | nfrimi 1505 | . 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
12 | 11 | nfrd 1500 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 Ⅎwnf 1440 [wsb 1742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 |
This theorem is referenced by: (None) |
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