Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfsb4t | GIF version |
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1888). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
Ref | Expression |
---|---|
nfsb4t | ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnf1 1436 | . . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑 | |
2 | 1 | nfal 1468 | . . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑 |
3 | nfnae 1610 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
4 | 2, 3 | nfan 1457 | . . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) |
5 | df-nf 1350 | . . . . . 6 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑)) | |
6 | 5 | albii 1359 | . . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 ↔ ∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑)) |
7 | hbsb4t 1889 | . . . . 5 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) | |
8 | 6, 7 | sylbi 114 | . . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) |
9 | 8 | imp 115 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)) |
10 | 4, 9 | nfd 1416 | . 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
11 | 10 | ex 108 | 1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∀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-in1 544 ax-in2 545 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-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 |
This theorem is referenced by: dvelimdf 1892 |
Copyright terms: Public domain | W3C validator |