Theorem mopick2 2109

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 1631. (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 2064	. . . 4 |- ( E* x ph -> A. x E* x ph )
2		hbe1 1495	. . . 4 |- ( E. x ( ph /\ ps ) -> A. x E. x ( ph /\ ps ) )
3	1, 2	hban 1547	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> A. x ( E* x ph /\ E. x ( ph /\ ps ) ) )
4		mopick 2104	. . . . . 6 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
5	4	ancld 325	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ( ph /\ ps ) ) )
6	5	anim1d 336	. . . 4 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ( ph /\ ps ) /\ ch ) ) )
7		df-3an 980	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
8	6, 7	imbitrrdi 162	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ph /\ ps /\ ch ) ) )
9	3, 8	eximdh 1611	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( E. x ( ph /\ ch ) -> E. x ( ph /\ ps /\ ch ) ) )
10	9	3impia 1200	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   E.wex 1492   E*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by: (None)