Theorem rmo4f

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006) (Revised by Thierry Arnoux, 11-Oct-2016) (Revised by Thierry Arnoux, 8-Mar-2017) (Revised by Thierry Arnoux, 8-Oct-2017)

	Ref	Expression
Hypotheses	rmo4f.1	|- F/_ x A
	rmo4f.2	|- F/_ y A
	rmo4f.3	|- F/ x ps
	rmo4f.4	|- ( x = y -> ( ph <-> ps ) )
Assertion	rmo4f	|- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )

Proof

Step	Hyp	Ref	Expression
1		rmo4f.1	|- F/_ x A
2		rmo4f.2	|- F/_ y A
3		rmo4f.3	|- F/ x ps
4		rmo4f.4	|- ( x = y -> ( ph <-> ps ) )
5		nfv	|- F/ y ph
6	1 2 5	rmo3f	|- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7	3 4	sbiev	|- ( [ y / x ] ph <-> ps )
8	7	anbi2i	|- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
9	8	imbi1i	|- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
10	9	2ralbii	|- ( A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
11	6 10	bitri	|- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )