Theorem mooran1 2586

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

Assertion

Ref	Expression
mooran1	⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))

Proof of Theorem mooran1

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	moimi 2576	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
3		moan 2583	. 2 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
4	2, 3	jaoi 871	1 ⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-mo 2570

This theorem is used by: (None)