Theorem r19.40 2651

Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.40	|- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )

Proof of Theorem r19.40

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	reximi 2594	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ph )
3		simpr 110	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	reximi 2594	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ps )
5	2, 4	jca 306	1 |- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-ral 2480  df-rex 2481

This theorem is referenced by:  rexanuz  11153  metequiv2  14732