Theorem mopick2 2024

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 1562. (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 1979	. . . 4 |- ( E* x ph -> A. x E* x ph )
2		hbe1 1424	. . . 4 |- ( E. x ( ph /\ ps ) -> A. x E. x ( ph /\ ps ) )
3	1, 2	hban 1479	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> A. x ( E* x ph /\ E. x ( ph /\ ps ) ) )
4		mopick 2019	. . . . . 6 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
5	4	ancld 318	. . . . 5 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ( ph /\ ps ) ) )
6	5	anim1d 329	. . . 4 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ( ph /\ ps ) /\ ch ) ) )
7		df-3an 921	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
8	6, 7	syl6ibr 160	. . 3 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ( ph /\ ch ) -> ( ph /\ ps /\ ch ) ) )
9	3, 8	eximdh 1542	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( E. x ( ph /\ ch ) -> E. x ( ph /\ ps /\ ch ) ) )
10	9	3impia 1135	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   E.wex 1421   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)