![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2205. (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 2203 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
6 | 1, 5 | mpd 15 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-ex 1774 df-nf 1778 |
This theorem is referenced by: exlimdd 2205 tz6.12cOLD 6912 ovmpodf 7560 gsum2d2lem 19893 2ndresdju 32383 padct 32451 stoweidlem27 45315 intsaluni 45617 isomenndlem 45818 tz6.12c-afv2 46522 |
Copyright terms: Public domain | W3C validator |