Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > alexdc | GIF version |
Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1536. (Contributed by Jim Kingdon, 2-Jun-2018.) |
Ref | Expression |
---|---|
alexdc | ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1434 | . . 3 ⊢ Ⅎ𝑥∀𝑥DECID 𝜑 | |
2 | notnotbdc 766 | . . . 4 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | |
3 | 2 | sps 1430 | . . 3 ⊢ (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) |
4 | 1, 3 | albid 1506 | . 2 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)) |
5 | alnex 1388 | . 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
6 | 4, 5 | syl6bb 185 | 1 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 DECID wdc 742 ∀wal 1241 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-fal 1249 df-nf 1350 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |