Theorem moanim 2529

Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Hypothesis

Ref	Expression
moanim.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
moanim	⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Proof of Theorem moanim

Step	Hyp	Ref	Expression
1		moanim.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		ibar 525	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mobid 2489	. . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
4	3	biimprcd 240	. 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃*𝑥𝜓))
5		simpl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
6	1, 5	exlimi 2086	. . . . 5 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
7		exmo 2495	. . . . . 6 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ∨ ∃*𝑥(𝜑 ∧ 𝜓))
8	7	ori 390	. . . . 5 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
9	6, 8	nsyl4 156	. . . 4 ⊢ (¬ ∃*𝑥(𝜑 ∧ 𝜓) → 𝜑)
10	9	con1i 144	. . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
11		moan 2524	. . 3 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
12	10, 11	ja 173	. 2 ⊢ ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
13	4, 12	impbii 199	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1704  Ⅎwnf 1708  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  moanimv  2531  moanmo  2532