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Mirrors > Home > ILE Home > Th. List > nnal | GIF version |
Description: The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.) |
Ref | Expression |
---|---|
nnal | ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnalim 1639 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | |
2 | 1 | con3i 627 | . 2 ⊢ (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) |
3 | alnex 1492 | . 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1346 ∃wex 1485 |
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 609 ax-in2 610 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 |
This theorem is referenced by: bj-stal 13784 |
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