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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 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 1537 ∃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 206 df-an 395 df-ex 1780 |
This theorem is referenced by: exnalimn 1844 equsexvw 2006 sbn 2274 ceqsex 3522 ceqsexv 3524 zfregs2 9730 ac6n 10482 nnunb 12472 alexsubALTlem3 23773 nmobndseqi 30299 bj-equsexvwd 35962 difunieq 36558 frege124d 42814 zfregs2VD 43904 |
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