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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 2641 | . . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
3 | 2 | moanim 2705 | . . 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 2631 | . 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 2620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-mo 2622 |
This theorem is referenced by: moaneu 2708 |
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