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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 2621. (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 2550 | . 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
4 | simpl 483 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
5 | 1, 4 | exlimi 2210 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
6 | 3, 5 | moanimlem 2620 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 Ⅎwnf 1786 ∃*wmo 2538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-nf 1787 df-mo 2540 |
This theorem is referenced by: moanmo 2624 reuxfrdf 30839 |
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