![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > alinexa | 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 | imnan 657 | . . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | 1 | albii 1400 | . 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) |
3 | alnex 1429 | . 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 182 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1283 ∃wex 1422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-5 1377 ax-gen 1379 ax-ie2 1424 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 |
This theorem is referenced by: sbnv 1811 ralnex 2363 |
Copyright terms: Public domain | W3C validator |