Theorem mopick2 2721

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 1886. (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		nfmo1 2640	. . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑
2		nfe1 2153	. . . 4 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
3	1, 2	nfan 1899	. . 3 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))
4		mopick 2709	. . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	4	ancld 553	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → (𝜑 ∧ 𝜓)))
6	5	anim1d 612	. . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∧ 𝜒)))
7		df-3an 1085	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
8	6, 7	syl6ibr 254	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → (𝜑 ∧ 𝜓 ∧ 𝜒)))
9	3, 8	eximd 2215	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)))
10	9	3impia 1113	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083  ∃wex 1779  ∃*wmo 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621

This theorem is referenced by:  moantr  35620