Theorem r19.40 3088

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 473	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	reximi 3011	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ph )
3		simpr 477	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	reximi 3011	. 2 |- ( E. x e. A ( ph /\ ps ) -> E. x e. A ps )
5	2, 4	jca 554	1 |- ( E. x e. A ( ph /\ ps ) -> ( E. x e. A ph /\ E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  rexanuz  14085  txflf  21810  metequiv2  22315  mzpcompact2lem  37314