Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > exlimdd | Structured version Visualization version GIF version |
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Ref | Expression |
---|---|
exlimdd.1 | ⊢ Ⅎ𝑥𝜑 |
exlimdd.2 | ⊢ Ⅎ𝑥𝜒 |
exlimdd.3 | ⊢ (𝜑 → ∃𝑥𝜓) |
exlimdd.4 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
exlimdd | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimdd.3 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | exlimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | exlimdd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
4 | exlimdd.4 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
5 | 4 | ex 449 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
6 | 2, 3, 5 | exlimd 2074 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
7 | 1, 6 | mpd 15 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: fvmptdf 6204 ovmpt2df 6690 ex-natded9.26 26668 exlimimdd 32367 suprnmpt 38350 stoweidlem43 38936 stoweidlem44 38937 stoweidlem54 38947 |
Copyright terms: Public domain | W3C validator |