Theorem mopick2 2271

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 1609. (Contributed by NM, 5-Apr-2004.) (Proof 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		nfmo1 2215	. . . 4 |- F/xE*xph
2		nfe1 1732	. . . 4 |- F/xE.x(ph /\ ps)
3	1, 2	nfan 1824	. . 3 |- F/x(E*xph /\ E.x(ph /\ ps))
4		mopick 2266	. . . . . 6 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
5	4	ancld 536	. . . . 5 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> (ph /\ ps)))
6	5	anim1d 547	. . . 4 |- ((E*xph /\ E.x(ph /\ ps)) -> ((ph /\ ch) -> ((ph /\ ps) /\ ch)))
7		df-3an 936	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
8	6, 7	syl6ibr 218	. . 3 |- ((E*xph /\ E.x(ph /\ ps)) -> ((ph /\ ch) -> (ph /\ ps /\ ch)))
9	3, 8	eximd 1770	. 2 |- ((E*xph /\ E.x(ph /\ ps)) -> (E.x(ph /\ ch) -> E.x(ph /\ ps /\ ch)))
10	9	3impia 1148	1 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)