![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > moim | GIF version |
Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
Ref | Expression |
---|---|
moim | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1522 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓) | |
2 | ax-4 1488 | . . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
3 | spsbim 1816 | . . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
4 | 2, 3 | anim12d 333 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) |
5 | 4 | imim1d 75 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
6 | 5 | alimdv 1852 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
7 | 1, 6 | alimd 1502 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
8 | ax-17 1507 | . . 3 ⊢ (𝜓 → ∀𝑦𝜓) | |
9 | 8 | mo3h 2053 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) |
10 | ax-17 1507 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
11 | 10 | mo3h 2053 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
12 | 7, 9, 11 | 3imtr4g 204 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 [wsb 1736 ∃*wmo 2001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 |
This theorem is referenced by: moimi 2065 euimmo 2067 moexexdc 2084 euexex 2085 rmoim 2889 rmoimi2 2891 ssrmof 3165 disjss1 3920 reusv1 4387 funmo 5146 uptx 12482 |
Copyright terms: Public domain | W3C validator |