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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 2620. (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 532 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | mobid 2549 | . 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
4 | simpl 486 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
5 | 1, 4 | exlimi 2215 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
6 | 3, 5 | moanimlem 2619 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 Ⅎwnf 1791 ∃*wmo 2537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-mo 2539 |
This theorem is referenced by: moanmo 2623 reuxfrdf 30558 |
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