Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > moanmo | Structured version Visualization version GIF version |
Description: Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.) |
Ref | Expression |
---|---|
moanmo | ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑) | |
2 | nfmo1 2637 | . . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
3 | 2 | moanim 2701 | . . 3 ⊢ (∃*𝑥(∃*𝑥𝜑 ∧ 𝜑) ↔ (∃*𝑥𝜑 → ∃*𝑥𝜑)) |
4 | 1, 3 | mpbir 233 | . 2 ⊢ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑) |
5 | ancom 463 | . . 3 ⊢ ((𝜑 ∧ ∃*𝑥𝜑) ↔ (∃*𝑥𝜑 ∧ 𝜑)) | |
6 | 5 | mobii 2627 | . 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) ↔ ∃*𝑥(∃*𝑥𝜑 ∧ 𝜑)) |
7 | 4, 6 | mpbir 233 | 1 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃*wmo 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2156 ax-12 2172 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-mo 2618 |
This theorem is referenced by: moaneu 2704 |
Copyright terms: Public domain | W3C validator |