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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. For a version requiring disjoint variables, but fewer axioms, see moanimv 2697. (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 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ibar 529 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | mobid 2627 | . 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
4 | simpl 483 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
5 | 1, 4 | exlimi 2207 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
6 | 3, 5 | moanimlem 2696 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 Ⅎwnf 1775 ∃*wmo 2613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-mo 2615 |
This theorem is referenced by: moanmo 2700 reuxfrdf 30182 |
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