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Theorem r19.29 2594
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
StepHypRef Expression
1 pm3.2 138 . . . 4  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
21ralimi 2520 . . 3  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ( ps  ->  ( ph  /\ 
ps ) ) )
3 rexim 2551 . . 3  |-  ( A. x  e.  A  ( ps  ->  ( ph  /\  ps ) )  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
42, 3syl 14 . 2  |-  ( A. x  e.  A  ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
54imp 123 1  |-  ( ( A. x  e.  A  ph 
/\  E. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wral 2435   E.wrex 2436
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-ral 2440  df-rex 2441
This theorem is referenced by:  r19.29r  2595  r19.29d2r  2601  r19.35-1  2607  triun  4077  ralxfrd  4424  elrnmptg  4840  fun11iun  5437  fmpt  5619  fliftfun  5748  epttop  12560  tgcnp  12679  lmtopcnp  12720  txlm  12749  metss  12964  bj-findis  13625
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