Theorem mo4f

Description: At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on y , ph can be replaced by the nonfreeness hypothesis |- F/ y ph with essentially the same proof. (Contributed by NM, 10-Apr-2004) Remove dependency on ax-13 . (Revised by Wolf Lammen, 19-Jan-2023)

	Ref	Expression
Hypotheses	mo4f.1	|- F/ x ps
	mo4f.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	mo4f	|- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Proof

Step	Hyp	Ref	Expression
1		mo4f.1	|- F/ x ps
2		mo4f.2	|- ( x = y -> ( ph <-> ps ) )
3		nfv	|- F/ y ph
4	3	mo3	|- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
5	1 2	sbiev	|- ( [ y / x ] ph <-> ps )
6	5	anbi2i	|- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
7	6	imbi1i	|- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
8	7	2albii	|- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
9	4 8	bitri	|- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )