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 2155 | . . 3 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | nfmov 2560 | . 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
3 | 19.8a 2182 | . . 3 ⊢ (𝜑 → ∃𝑦𝜑) | |
4 | 3 | moimi 2545 | . 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑) |
5 | 2, 4 | alrimi 2215 | 1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 ∃*wmo 2538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-mo 2540 |
This theorem is referenced by: 2eu5 2658 |
Copyright terms: Public domain | W3C validator |