Theorem moanim 2621

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2620. (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 532	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mobid 2549	. 2 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓)))
4		simpl 486	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
5	1, 4	exlimi 2215	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑)
6	3, 5	moanimlem 2619	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1791  ∃*wmo 2537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-mo 2539

This theorem is referenced by:  moanmo  2623  reuxfrdf  30558