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Mirrors > Home > MPE Home > Th. List > moanim | Structured version Visualization version GIF version |
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
Ref | Expression |
---|---|
moanim.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
moanim | ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moanim.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | ibar 524 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | mobid 2477 | . . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
4 | 3 | biimprcd 239 | . 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃*𝑥𝜓)) |
5 | simpl 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
6 | 1, 5 | exlimi 2073 | . . . . 5 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
7 | exmo 2483 | . . . . . 6 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ∨ ∃*𝑥(𝜑 ∧ 𝜓)) | |
8 | 7 | ori 389 | . . . . 5 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) |
9 | 6, 8 | nsyl4 155 | . . . 4 ⊢ (¬ ∃*𝑥(𝜑 ∧ 𝜓) → 𝜑) |
10 | 9 | con1i 143 | . . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓)) |
11 | moan 2512 | . . 3 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓)) | |
12 | 10, 11 | ja 172 | . 2 ⊢ ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) |
13 | 4, 12 | impbii 198 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃wex 1695 Ⅎwnf 1699 ∃*wmo 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-mo 2463 |
This theorem is referenced by: moanimv 2519 moanmo 2520 |
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