| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 2moexv | Structured version Visualization version GIF version | ||
| Description: Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.) |
| Ref | Expression |
|---|---|
| 2moexv | ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
| 2 | 1 | nfmov 2560 | . 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
| 3 | 19.8a 2181 | . . 3 ⊢ (𝜑 → ∃𝑦𝜑) | |
| 4 | 3 | moimi 2545 | . 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑) |
| 5 | 2, 4 | alrimi 2213 | 1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 |
| This theorem is referenced by: 2eu5 2656 |
| Copyright terms: Public domain | W3C validator |