Theorem mooran2 2099

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 2097	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
2		olc 711	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
3	2	moimi 2091	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜓)
4	1, 3	jca 306	1 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708  ∃*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by: (None)