Theorem r19.29 2567

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		pm3.2 138	. . . 4 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi 2493	. . 3 |- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2524	. . 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 123	1 |- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2414   E.wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-ral 2419  df-rex 2420

This theorem is referenced by:  r19.29r  2568  r19.29d2r  2574  r19.35-1  2579  triun  4034  ralxfrd  4378  elrnmptg  4786  fun11iun  5381  fmpt  5563  fliftfun  5690  epttop  12248  tgcnp  12367  lmtopcnp  12408  txlm  12437  metss  12652  bj-findis  13166