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Metamath Proof Explorer

Theorem eqreu

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 eqreu.1 
 |-  ( x = B -> ( ph <-> ps ) )
2 ralbiim 
 |-  ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) )
3 1 ceqsralv 
 |-  ( B e. A -> ( A. x e. A ( x = B -> ph ) <-> ps ) )
4 3 anbi2d 
 |-  ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
5 2 4 bitrid 
 |-  ( B e. A -> ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
6 reu6i 
 |-  ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
7 6 ex 
 |-  ( B e. A -> ( A. x e. A ( ph <-> x = B ) -> E! x e. A ph ) )
8 5 7 sylbird 
 |-  ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph ) )
9 8 3impib 
 |-  ( ( B e. A /\ A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph )
10 9 3com23 
 |-  ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )