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Mirrors > Home > MPE Home > Th. List > exlimimdd | Structured version Visualization version GIF version |
Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2210. (Revised by Wolf Lammen, 3-Sep-2023.) |
Ref | Expression |
---|---|
exlimdd.1 | ⊢ Ⅎ𝑥𝜑 |
exlimdd.2 | ⊢ Ⅎ𝑥𝜒 |
exlimdd.3 | ⊢ (𝜑 → ∃𝑥𝜓) |
exlimimdd.4 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
exlimimdd | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimdd.3 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | exlimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | exlimdd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
4 | exlimimdd.4 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
5 | 2, 3, 4 | exlimd 2208 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
6 | 1, 5 | mpd 15 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: exlimdd 2210 tz6.12c 6688 ovmpodf 7295 gsum2d2lem 19022 padct 30381 stoweidlem27 42189 intsaluni 42489 isomenndlem 42689 tz6.12c-afv2 43318 |
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