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Metamath Proof Explorer

Theorem eqeu

Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009)

Ref Expression
Hypothesis eqeu.1 
|- ( x = A -> ( ph <-> ps ) )
Assertion eqeu 
|- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Proof

Step Hyp Ref Expression
1 eqeu.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 1 spcegv 
 |-  ( A e. B -> ( ps -> E. x ph ) )
3 2 imp 
 |-  ( ( A e. B /\ ps ) -> E. x ph )
4 3 3adant3 
 |-  ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5 eqeq2 
 |-  ( y = A -> ( x = y <-> x = A ) )
6 5 imbi2d 
 |-  ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7 6 albidv 
 |-  ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8 7 spcegv 
 |-  ( A e. B -> ( A. x ( ph -> x = A ) -> E. y A. x ( ph -> x = y ) ) )
9 8 imp 
 |-  ( ( A e. B /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
10 9 3adant2 
 |-  ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11 eu3v 
 |-  ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
12 4 10 11 sylanbrc 
 |-  ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )