![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 19.37-1 | GIF version |
Description: One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.) |
Ref | Expression |
---|---|
19.37-1.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.37-1 | ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.37-1.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | 19.3 1498 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
3 | 19.35-1 1567 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
4 | 2, 3 | syl5bir 152 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1294 Ⅎwnf 1401 ∃wex 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-4 1452 ax-ial 1479 |
This theorem depends on definitions: df-bi 116 df-nf 1402 |
This theorem is referenced by: 19.37aiv 1617 spcimegft 2711 eqvincg 2755 |
Copyright terms: Public domain | W3C validator |