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Mirrors > Home > MPE Home > Th. List > hbsbw | Structured version Visualization version GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2529 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 23-May-2024.) |
Ref | Expression |
---|---|
hbsbw.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
hbsbw | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2068 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
2 | hbsbw.1 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑧𝜑) | |
3 | 2 | imim2i 16 | . . . . . . 7 ⊢ ((𝑥 = 𝑤 → 𝜑) → (𝑥 = 𝑤 → ∀𝑧𝜑)) |
4 | 3 | alimi 1814 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑)) |
5 | 19.21v 1942 | . . . . . . . 8 ⊢ (∀𝑧(𝑥 = 𝑤 → 𝜑) ↔ (𝑥 = 𝑤 → ∀𝑧𝜑)) | |
6 | 5 | biimpri 227 | . . . . . . 7 ⊢ ((𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑧(𝑥 = 𝑤 → 𝜑)) |
7 | 6 | alimi 1814 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑥∀𝑧(𝑥 = 𝑤 → 𝜑)) |
8 | ax-11 2154 | . . . . . 6 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑤 → 𝜑) → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)) | |
9 | 4, 7, 8 | 3syl 18 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)) |
10 | 9 | imim2i 16 | . . . 4 ⊢ ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → (𝑤 = 𝑦 → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑))) |
11 | 19.21v 1942 | . . . 4 ⊢ (∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ (𝑤 = 𝑦 → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
12 | 10, 11 | sylibr 233 | . . 3 ⊢ ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → ∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) |
13 | 12 | hbal 2167 | . 2 ⊢ (∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → ∀𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) |
14 | 1, 13 | hbxfrbi 1827 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-11 2154 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-sb 2068 |
This theorem is referenced by: nfsbv 2324 hbab 2726 hblem 2870 |
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