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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 1646 | . . 3 ⊢ (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑) | |
2 | 1 | alimi 1455 | . 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑) |
3 | alnex 1499 | . 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) | |
4 | 2, 3 | sylib 122 | 1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1351 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 |
This theorem is referenced by: nalset 4132 bj-nalset 14507 |
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