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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 2672. (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 2589 | . 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
4 | simpl 483 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
5 | 1, 4 | exlimi 2182 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
6 | 3, 5 | moanimlem 2671 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 Ⅎwnf 1765 ∃*wmo 2574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-12 2141 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-nf 1766 df-mo 2576 |
This theorem is referenced by: moanimvOLD 2675 moanmo 2676 reuxfrdf 29947 |
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