Theorem mooran2 2582

Description: "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
mooran2	⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Proof of Theorem mooran2

Step	Hyp	Ref	Expression
1		moor 2580	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
2		olc 882	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
3	2	moimi 2571	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜓)
4	1, 3	jca 521	1 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∃*wmo 2563

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-mo 2565

This theorem is used by:  rmoun  33083