Theorem r19.29 2631

Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.29	|- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Proof of Theorem r19.29

Step	Hyp	Ref	Expression
1		pm3.2 139	. . . 4 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi 2557	. . 3 |- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2588	. . 3 |- ( A. x e. A ( ps -> ( ph /\ ps ) ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
4	2, 3	syl 14	. 2 |- ( A. x e. A ph -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
5	4	imp 124	1 |- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2472   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-ral 2477  df-rex 2478

This theorem is referenced by:  r19.29r  2632  r19.29d2r  2638  r19.35-1  2644  triun  4140  ralxfrd  4493  elrnmptg  4914  fun11iun  5521  fmpt  5708  fliftfun  5839  rhmdvdsr  13671  epttop  14258  tgcnp  14377  lmtopcnp  14418  txlm  14447  metss  14662  bj-findis  15471