Theorem moanim 2610

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2609. (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 528	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mobid 2538	. 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
4		simpl 482	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
5	1, 4	exlimi 2202	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
6	3, 5	moanimlem 2608	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  Ⅎwnf 1777  ∃*wmo 2526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-mo 2528

This theorem is referenced by:  moanmo  2612  reuxfrdf  32236