![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 19.32r | GIF version |
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1610. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Ref | Expression |
---|---|
19.32r.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.32r | ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.32r.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | orc 666 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | alrimi 1456 | . 2 ⊢ (𝜑 → ∀𝑥(𝜑 ∨ 𝜓)) |
4 | olc 665 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
5 | 4 | alimi 1385 | . 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓)) |
6 | 3, 5 | jaoi 669 | 1 ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 662 ∀wal 1283 Ⅎwnf 1390 |
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-io 663 ax-5 1377 ax-gen 1379 ax-4 1441 |
This theorem depends on definitions: df-bi 115 df-nf 1391 |
This theorem is referenced by: 19.31r 1612 |
Copyright terms: Public domain | W3C validator |