Theorem rmoi

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012) (Revised by NM, 16-Jun-2017)

	Ref	Expression
Hypotheses	rmoi.b	|- ( x = B -> ( ph <-> ps ) )
	rmoi.c	|- ( x = C -> ( ph <-> ch ) )
Assertion	rmoi	|- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )

Proof

Step	Hyp	Ref	Expression
1		rmoi.b	|- ( x = B -> ( ph <-> ps ) )
2		rmoi.c	|- ( x = C -> ( ph <-> ch ) )
3	1 2	rmob	|- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
4	3	biimp3ar	|- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )