![]() |
Mathbox for ML |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > exellim | Structured version Visualization version GIF version |
Description: Closed form of exellimddv 33530. See also exlimim 33526 for a more general theorem. (Contributed by ML, 17-Jul-2020.) |
Ref | Expression |
---|---|
exellim | ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2184 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑) | |
2 | nfv 1995 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | sp 2207 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑)) | |
4 | 1, 2, 3 | exlimd 2243 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
5 | 4 | impcom 394 | 1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1629 ∃wex 1852 ∈ wcel 2145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-ex 1853 df-nf 1858 |
This theorem is referenced by: exellimddv 33530 |
Copyright terms: Public domain | W3C validator |