Theorem r19.29 2688

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 2613	. . 3 |- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2644	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wral 2528   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-ral 2533  df-rex 2534

This theorem is used by:  r19.29r  2689  r19.29d2r  2695  r19.35-1  2701  triun  4242  ralxfrd  4608  elrnmptg  5034  fun11iun  5660  fmpt  5858  fliftfun  6002  rhmdvdsr  14482  epttop  15191  tgcnp  15310  lmtopcnp  15351  txlm  15380  metss  15595  bj-findis  17005