Theorem eu4

Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995)

	Ref	Expression
Hypothesis	eu4.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	eu4	|- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )

Proof

Step	Hyp	Ref	Expression
1		eu4.1	|- ( x = y -> ( ph <-> ps ) )
2		df-eu	|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
3	1	mo4	|- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i	|- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
5	2 4	bitri	|- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )