Theorem mooran2 2073

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 2071	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
2		olc 701	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
3	2	moimi 2065	. 2 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜓)
4	1, 3	jca 304	1 ⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  ∃*wmo 2001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by: (None)