Metamath Proof Explorer


Theorem mopick

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)

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

Proof

Step Hyp Ref Expression
1 df-mo
 |-  ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2 sp
 |-  ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
3 pm3.45
 |-  ( ( ph -> x = y ) -> ( ( ph /\ ps ) -> ( x = y /\ ps ) ) )
4 3 aleximi
 |-  ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> E. x ( x = y /\ ps ) ) )
5 ax12ev2
 |-  ( E. x ( x = y /\ ps ) -> ( x = y -> ps ) )
6 4 5 syl6
 |-  ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( x = y -> ps ) ) )
7 2 6 syl5d
 |-  ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
8 7 exlimiv
 |-  ( E. y A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
9 1 8 sylbi
 |-  ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
10 9 imp
 |-  ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )