Theorem mooran1 2013

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 107	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	moimi 2006	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜑 ∧ 𝜓))
3		moan 2010	. 2 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓))
4	2, 3	jaoi 668	1 ⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)