Theorem r19.29 2634

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 2560	. . 3 |- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2591	. . 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 2475   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:  r19.29r  2635  r19.29d2r  2641  r19.35-1  2647  triun  4144  ralxfrd  4497  elrnmptg  4918  fun11iun  5525  fmpt  5712  fliftfun  5843  rhmdvdsr  13731  epttop  14326  tgcnp  14445  lmtopcnp  14486  txlm  14515  metss  14730  bj-findis  15625