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Mirrors > Home > ILE Home > Th. List > alexnim | GIF version |
Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Ref | Expression |
---|---|
alexnim | ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnalim 1625 | . . 3 ⊢ (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑) | |
2 | 1 | alimi 1431 | . 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑) |
3 | alnex 1475 | . 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) | |
4 | 2, 3 | sylib 121 | 1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1329 ∃wex 1468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 |
This theorem is referenced by: nalset 4058 bj-nalset 13093 |
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