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Metamath Proof Explorer

Theorem r19.44v

Description: One direction of a restricted quantifier version of 19.44 . The other direction holds when A is nonempty, see r19.44zv . (Contributed by NM, 2-Apr-2004)

		Ref	Expression
	Assertion	r19.44v	\|- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ 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	\|- ( ps -> ps )
3	2	rexlimivw	\|- ( E. x e. A ps -> ps )
4	3	orim2i	\|- ( ( E. x e. A ph \/ E. x e. A ps ) -> ( E. x e. A ph \/ ps ) )
5	1 4	sylbi	\|- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) )