![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > exanaliim | GIF version |
Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Ref | Expression |
---|---|
exanaliim | ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annimim 816 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) | |
2 | 1 | eximi 1532 | . 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑 → 𝜓)) |
3 | exnalim 1578 | . 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∀wal 1283 ∃wex 1422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-4 1441 ax-17 1460 ax-ial 1468 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 df-nf 1391 |
This theorem is referenced by: rexnalim 2364 nssr 3068 nssssr 4012 brprcneu 5244 |
Copyright terms: Public domain | W3C validator |