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Mirrors > Home > MPE Home > Th. List > nexdh | Structured version Visualization version GIF version |
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
nexdh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
nexdh.2 | ⊢ (𝜑 → ¬ 𝜓) |
Ref | Expression |
---|---|
nexdh | ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nexdh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | nexdh.2 | . . 3 ⊢ (𝜑 → ¬ 𝜓) | |
3 | 1, 2 | alrimih 1805 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝜓) |
4 | alnex 1763 | . 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
5 | 3, 4 | sylib 219 | 1 ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1520 ∃wex 1761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 |
This theorem depends on definitions: df-bi 208 df-ex 1762 |
This theorem is referenced by: nexdv 1914 nexd 2188 |
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