Theorem mopick2 2097

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 1619. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Proof of Theorem mopick2

Step	Hyp	Ref	Expression
1		hbmo1 2052	. . . 4 |- ( E* x ph -> A. x E* x ph )
2		hbe1 1483	. . . 4 |- ( E. x ( ph /\ ps ) -> A. x E. x ( ph /\ ps ) )
3	1, 2	hban 1535	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> A. x ( E* x ph /\ E. x ( ph /\ ps ) ) )
4		mopick 2092	. . . . . 6 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
5	4	ancld 323	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ( ph /\ ps ) ) )
6	5	anim1d 334	. . . 4 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ( ph /\ ps ) /\ ch ) ) )
7		df-3an 970	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
8	6, 7	syl6ibr 161	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ph /\ ps /\ ch ) ) )
9	3, 8	eximdh 1599	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( E. x ( ph /\ ch ) -> E. x ( ph /\ ps /\ ch ) ) )
10	9	3impia 1190	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   E.wex 1480   E*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by: (None)