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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0limd | Structured version Visualization version GIF version |
Description: Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
Ref | Expression |
---|---|
n0limd.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
n0limd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
Ref | Expression |
---|---|
n0limd | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0limd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
2 | n0 4358 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
4 | n0limd.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
5 | 3, 4 | exlimddv 1932 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-ne 2938 df-dif 3965 df-nul 4339 |
This theorem is referenced by: dimlssid 33659 |
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