Theorem mopick 2307

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 2245	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		sp 1883	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
3		pm3.45 835	. . . . . . 7 |- ( ( ph -> x = y ) -> ( ( ph /\ ps ) -> ( x = y /\ ps ) ) )
4	3	aleximi 1674	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> E. x ( x = y /\ ps ) ) )
5		sb56 2196	. . . . . . 7 |- ( E. x ( x = y /\ ps ) <-> A. x ( x = y -> ps ) )
6		sp 1883	. . . . . . 7 |- ( A. x ( x = y -> ps ) -> ( x = y -> ps ) )
7	5, 6	sylbi 195	. . . . . 6 |- ( E. x ( x = y /\ ps ) -> ( x = y -> ps ) )
8	4, 7	syl6 31	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( x = y -> ps ) ) )
9	2, 8	syl5d 66	. . . 4 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
10	9	exlimiv 1743	. . 3 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
11	1, 10	sylbi 195	. 2 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	11	imp 427	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 367   A.wal 1403   E.wex 1633   E*wmo 2239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243

This theorem is referenced by:  eupick  2309  mopick2  2313  moexex  2314  morex  3232  imadif  5600  cmetss  21937