Theorem prtlem18

Description: Lemma for prter2 . (Contributed by Rodolfo Medina, 15-Oct-2010) (Revised by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Hypothesis	prtlem18.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
Assertion	prtlem18	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v <-> z .~ w ) ) )

Proof

Step	Hyp	Ref	Expression
1		prtlem18.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
2		rspe	|- ( ( v e. A /\ ( z e. v /\ w e. v ) ) -> E. v e. A ( z e. v /\ w e. v ) )
3	2	expr	|- ( ( v e. A /\ z e. v ) -> ( w e. v -> E. v e. A ( z e. v /\ w e. v ) ) )
4	1	prtlem13	|- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )
5	3 4	syl6ibr	|- ( ( v e. A /\ z e. v ) -> ( w e. v -> z .~ w ) )
6	5	a1i	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v -> z .~ w ) ) )
7	1	prtlem13	|- ( z .~ w <-> E. p e. A ( z e. p /\ w e. p ) )
8		prtlem17	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( E. p e. A ( z e. p /\ w e. p ) -> w e. v ) ) )
9	7 8	syl7bi	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( z .~ w -> w e. v ) ) )
10	6 9	impbidd	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v <-> z .~ w ) ) )