Theorem mopick2 2024

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 1562. (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 1979	. . . 4 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
2		hbe1 1424	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥∃𝑥(𝜑 ∧ 𝜓))
3	1, 2	hban 1479	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)))
4		mopick 2019	. . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	4	ancld 318	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → (𝜑 ∧ 𝜓)))
6	5	anim1d 329	. . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∧ 𝜒)))
7		df-3an 921	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
8	6, 7	syl6ibr 160	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → (𝜑 ∧ 𝜓 ∧ 𝜒)))
9	3, 8	eximdh 1542	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)))
10	9	3impia 1135	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919  ∃wex 1421  ∃*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-3an 921  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)