Theorem mopick2 1837

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 1447. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
mopick2	|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Proof of Theorem mopick2

Step	Hyp	Ref	Expression
1		hbmo1 1802	. . . 4 |- (E*xph -> A.xE*xph)
2		hbe1 1363	. . . 4 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
3	1, 2	hban 1356	. . 3 |- ((E*xph /\ E.x(ph /\ ps)) -> A.x(E*xph /\ E.x(ph /\ ps)))
4		mopick 1833	. . . . . 6 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
5	4	ancld 322	. . . . 5 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> (ph /\ ps)))
6	5	anim1d 619	. . . 4 |- ((E*xph /\ E.x(ph /\ ps)) -> ((ph /\ ch) -> ((ph /\ ps) /\ ch)))
7		df-3an 860	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
8	6, 7	syl6ibr 230	. . 3 |- ((E*xph /\ E.x(ph /\ ps)) -> ((ph /\ ch) -> (ph /\ ps /\ ch)))
9	3, 8	eximd 1410	. 2 |- ((E*xph /\ E.x(ph /\ ps)) -> (E.x(ph /\ ch) -> E.x(ph /\ ps /\ ch)))
10	9	3impia 1064	1 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  E.wex 1326  E*wmo 1772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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