ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexlimd Unicode version

Theorem rexlimd 2544
Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
rexlimd.1  |-  F/ x ph
rexlimd.2  |-  F/ x ch
rexlimd.3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
rexlimd  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )

Proof of Theorem rexlimd
StepHypRef Expression
1 rexlimd.1 . . 3  |-  F/ x ph
2 rexlimd.3 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
31, 2ralrimi 2501 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd.2 . . 3  |-  F/ x ch
54r19.23 2538 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) )
63, 5sylib 121 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1436    e. wcel 1480   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  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420
This theorem is referenced by:  rexlimdv  2546  ralxfrALT  4383  fvmptt  5505  ffnfv  5571  nneneq  6744  ac6sfi  6785  prarloclem3step  7297  prmuloc2  7368  caucvgprprlemaddq  7509  axpre-suploclemres  7702  lbzbi  9401  divalglemeunn  11603  divalglemeuneg  11605  oddpwdclemdvds  11833  oddpwdclemndvds  11834
  Copyright terms: Public domain W3C validator