Theorem moanim 2651

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2650. (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 538	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mobid 2581	. 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
4		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
5	1, 4	exlimi 2256	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
6	3, 5	moanimlem 2649	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  Ⅎwnf 1816  ∃*wmo 2568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2570

This theorem is used by:  moanmo  2653  reuxfrdf  32874