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Mirrors > Home > MPE Home > Th. List > moaneu | Structured version Visualization version GIF version |
Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Ref | Expression |
---|---|
moaneu | ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moanmo 2684 | . 2 ⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) | |
2 | eumo 2638 | . . . 4 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
3 | 2 | anim2i 619 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) → (𝜑 ∧ ∃*𝑥𝜑)) |
4 | 3 | moimi 2603 | . 2 ⊢ (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) → ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃*wmo 2596 ∃!weu 2628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-mo 2598 df-eu 2629 |
This theorem is referenced by: (None) |
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