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| Mirrors > Home > MPE Home > Th. List > alinexa | Structured version Visualization version GIF version | ||
| Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |
| Ref | Expression |
|---|---|
| alinexa | ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imnang 1842 | . 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) | |
| 2 | alnex 1781 | . 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: exnalimn 1844 equsexvw 2004 sbn 2280 ceqsex 3530 ceqsexv 3532 zfregs2 9773 ac6n 10525 nnunb 12522 alexsubALTlem3 24057 nmobndseqi 30798 bj-equsexvwd 36782 difunieq 37375 frege124d 43774 zfregs2VD 44861 |
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