Theorem moanim 2646

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2645. (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	. . 3 ⊢ Ⅎ𝑥𝜑
2		ibar 536	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mobid 2576	. 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
4		simpl 486	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
5	1, 4	exlimi 2251	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
6	3, 5	moanimlem 2644	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  Ⅎwnf 1802  ∃*wmo 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803  df-mo 2565

This theorem is referenced by:  moanmo  2648  reuxfrdf  32648