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Mirrors > Home > ILE Home > Th. List > nexd | GIF version |
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
nexd.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
nexd.2 | ⊢ (𝜑 → ¬ 𝜓) |
Ref | Expression |
---|---|
nexd | ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nexd.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | nexd.2 | . . 3 ⊢ (𝜑 → ¬ 𝜓) | |
3 | 1, 2 | alrimih 1449 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝜓) |
4 | alnex 1479 | . 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
5 | 3, 4 | sylib 121 | 1 ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1333 ∃wex 1472 |
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 604 ax-in2 605 ax-5 1427 ax-gen 1429 ax-ie2 1474 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 |
This theorem is referenced by: nexdv 1916 |
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