![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eupickbi | GIF version |
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
eupickbi | ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupicka 1980 | . . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) | |
2 | 1 | ex 108 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
3 | hba1 1433 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓)) | |
4 | ancl 301 | . . . . . . 7 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓))) | |
5 | simpl 102 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
6 | 4, 5 | impbid1 130 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 ↔ (𝜑 ∧ 𝜓))) |
7 | 6 | sps 1430 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 ↔ (𝜑 ∧ 𝜓))) |
8 | 3, 7 | eubidh 1906 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥(𝜑 ∧ 𝜓))) |
9 | euex 1930 | . . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
10 | 8, 9 | syl6bi 152 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
11 | 10 | com12 27 | . 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
12 | 2, 11 | impbid 120 | 1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 ∃!weu 1900 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |