Theorem r19.40 2585

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 108	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	reximi 2529	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ph )
3		simpr 109	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	reximi 2529	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ps )
5	2, 4	jca 304	1 |- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-ral 2421  df-rex 2422

This theorem is referenced by:  rexanuz  10760  metequiv2  12665