Theorem mopick2 2121

Description: "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1642. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
mopick2	⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem mopick2

Step	Hyp	Ref	Expression
1		hbmo1 2076	. . . 4 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
2		hbe1 1506	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥∃𝑥(𝜑 ∧ 𝜓))
3	1, 2	hban 1558	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)))
4		mopick 2116	. . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	4	ancld 325	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → (𝜑 ∧ 𝜓)))
6	5	anim1d 336	. . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∧ 𝜒)))
7		df-3an 982	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
8	6, 7	imbitrrdi 162	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → (𝜑 ∧ 𝜓 ∧ 𝜒)))
9	3, 8	eximdh 1622	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)))
10	9	3impia 1202	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980  ∃wex 1503  ∃*wmo 2039

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042

This theorem is referenced by: (None)