![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > moim | Structured version Visualization version GIF version |
Description: "At most one" reverses implication. (Contributed by NM, 22-Apr-1995.) |
Ref | Expression |
---|---|
moim | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imim1 83 | . . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))) | |
2 | 1 | al2imi 1891 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
3 | 2 | eximdv 1998 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | mo2v 2625 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)) | |
5 | mo2v 2625 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
6 | 3, 4, 5 | 3imtr4g 285 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1629 ∃wex 1852 ∃*wmo 2619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ex 1853 df-nf 1858 df-eu 2622 df-mo 2623 |
This theorem is referenced by: moimi 2669 euimmo 2671 moexex 2690 rmoim 3559 rmoimi2 3561 disjss1 4760 disjss3 4785 funmo 6047 brdom6disj 9556 uptx 21649 taylf 24335 moimd 29666 ssrmo 29673 funressnfv 41728 |
Copyright terms: Public domain | W3C validator |