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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 ) )