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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 1836 | . 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) | |
2 | alnex 1775 | . 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | |
3 | 1, 2 | bitri 274 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 |
This theorem is referenced by: exnalimn 1838 equsexvw 2000 sbn 2269 ceqsex 3512 ceqsexv 3514 zfregs2 9758 ac6n 10510 nnunb 12501 alexsubALTlem3 23997 nmobndseqi 30661 bj-equsexvwd 36386 difunieq 36981 frege124d 43330 zfregs2VD 44419 |
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