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Mirrors > Home > ILE Home > Th. List > nnral | GIF version |
Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1649. (Contributed by Jim Kingdon, 1-Aug-2024.) |
Ref | Expression |
---|---|
nnral | ⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexnalim 2466 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑) | |
2 | 1 | con3i 632 | . 2 ⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) |
3 | ralnex 2465 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑) | |
4 | 2, 3 | sylibr 134 | 1 ⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wral 2455 ∃wrex 2456 |
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 df-ral 2460 df-rex 2461 |
This theorem is referenced by: onntri13 7237 onntri24 7241 |
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