Theorem r19.45v

Description: Restricted quantifier version of one direction of 19.45 . The other direction holds when A is nonempty, see r19.45zv . (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.45v	|- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) )

Proof

Step	Hyp	Ref	Expression
1		r19.43	|- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		id	|- ( ph -> ph )
3	2	rexlimivw	|- ( E. x e. A ph -> ph )
4	3	orim1i	|- ( ( E. x e. A ph \/ E. x e. A ps ) -> ( ph \/ E. x e. A ps ) )
5	1 4	sylbi	|- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) )