Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 19.30dc | GIF version |
Description: Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Ref | Expression |
---|---|
19.30dc | ⊢ (DECID ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 776 | . 2 ⊢ (DECID ∃𝑥𝜓 ↔ (∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓)) | |
2 | olc 664 | . . . 4 ⊢ (∃𝑥𝜓 → (∀𝑥𝜑 ∨ ∃𝑥𝜓)) | |
3 | 2 | a1d 22 | . . 3 ⊢ (∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) |
4 | alnex 1428 | . . . . 5 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
5 | orel2 677 | . . . . . 6 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑)) | |
6 | 5 | al2imi 1387 | . . . . 5 ⊢ (∀𝑥 ¬ 𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → ∀𝑥𝜑)) |
7 | 4, 6 | sylbir 133 | . . . 4 ⊢ (¬ ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → ∀𝑥𝜑)) |
8 | orc 665 | . . . 4 ⊢ (∀𝑥𝜑 → (∀𝑥𝜑 ∨ ∃𝑥𝜓)) | |
9 | 7, 8 | syl6 33 | . . 3 ⊢ (¬ ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) |
10 | 3, 9 | jaoi 668 | . 2 ⊢ ((∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) |
11 | 1, 10 | sylbi 119 | 1 ⊢ (DECID ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 661 DECID wdc 775 ∀wal 1282 ∃wex 1421 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-gen 1378 ax-ie2 1423 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-tru 1287 df-fal 1290 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |