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Mirrors > Home > ILE Home > Th. List > hbsbd | GIF version |
Description: Deduction version of hbsb 1942. (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 1455 | . . 3 ⊢ Ⅎ𝑧𝜑 |
3 | hbsbd.3 | . . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) | |
4 | 1, 3 | nfdh 1517 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) |
5 | 2, 4 | nfim1 1564 | . . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓) |
6 | 5 | nfsb 1939 | . . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) |
7 | hbsbd.1 | . . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑) | |
8 | 7 | sbrim 1949 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
9 | 8 | nfbii 1466 | . . . 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 1518 | . 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
12 | 11 | nfrd 1513 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 Ⅎwnf 1453 [wsb 1755 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 |
This theorem is referenced by: (None) |
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