Theorem mopick 2361

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)

Assertion

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

Proof of Theorem mopick

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2282	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		sp 1808	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
3		id 22	. . . . . . . 8 |- ( ( ph -> x = y ) -> ( ph -> x = y ) )
4	3	anim1d 564	. . . . . . 7 |- ( ( ph -> x = y ) -> ( ( ph /\ ps ) -> ( x = y /\ ps ) ) )
5	4	aleximi 1632	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> E. x ( x = y /\ ps ) ) )
6		sb56 2154	. . . . . . 7 |- ( E. x ( x = y /\ ps ) <-> A. x ( x = y -> ps ) )
7		sp 1808	. . . . . . 7 |- ( A. x ( x = y -> ps ) -> ( x = y -> ps ) )
8	6, 7	sylbi 195	. . . . . 6 |- ( E. x ( x = y /\ ps ) -> ( x = y -> ps ) )
9	5, 8	syl6 33	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( x = y -> ps ) ) )
10	2, 9	syl5d 67	. . . 4 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
11	10	exlimiv 1698	. . 3 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	1, 11	sylbi 195	. 2 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
13	12	imp 429	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1377   E.wex 1596   E*wmo 2276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280

This theorem is referenced by:  eupick  2364  mopick2  2370  moexex  2371  moexexOLD  2372  morex  3287  imadif  5661  cmetss  21488