Theorem r19.40 2464

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 102	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	reximi 2416	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ph )
3		simpr 103	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	reximi 2416	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ps )
5	2, 4	jca 290	1 |- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 97   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312

This theorem is referenced by:  rexanuz  9587