Theorem r19.29 2495

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		pm3.2 137	. . . 4 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi 2427	. . 3 |- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2456	. . 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 122	1 |- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wral 2349   E.wrex 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-ral 2354  df-rex 2355

This theorem is referenced by:  r19.29r  2496  r19.29d2r  2500  r19.35-1  2505  triun  3896  ralxfrd  4220  elrnmptg  4614  fun11iun  5178  fmpt  5351  fliftfun  5467  bj-findis  10932