Theorem mooran1 2555

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 482	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	moimi 2545	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
3		moan 2552	. 2 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
4	2, 3	jaoi 857	1 ⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-mo 2540

This theorem is referenced by: (None)