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Mirrors > Home > ILE Home > Th. List > hbsb4t | 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.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
hbsb4t | ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 1433 | . . 3 ⊢ (∀𝑧𝜑 → ∀𝑧∀𝑧𝜑) | |
2 | 1 | hbsb4 1888 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑)) |
3 | spsbim 1724 | . . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑)) | |
4 | 3 | sps 1430 | . . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑)) |
5 | ax-4 1400 | . . . . . . 7 ⊢ (∀𝑧𝜑 → 𝜑) | |
6 | 5 | sbimi 1647 | . . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 → [𝑦 / 𝑥]𝜑) |
7 | 6 | alimi 1344 | . . . . 5 ⊢ (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
8 | 7 | a1i 9 | . . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)) |
9 | 4, 8 | imim12d 68 | . . 3 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) |
10 | 9 | a7s 1343 | . 2 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) |
11 | 2, 10 | syl5 28 | 1 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 [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-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-nf 1350 df-sb 1646 |
This theorem is referenced by: nfsb4t 1890 |
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