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

Theorem rexlimd 2487
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 2445 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd.2 . . 3  |-  F/ x ch
54r19.23 2481 . 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 1395    e. wcel 1439   A.wral 2360   E.wrex 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-ral 2365  df-rex 2366
This theorem is referenced by:  rexlimdv  2489  ralxfrALT  4302  fvmptt  5407  ffnfv  5470  nneneq  6627  ac6sfi  6668  prarloclem3step  7109  prmuloc2  7180  caucvgprprlemaddq  7321  lbzbi  9155  divalglemeunn  11253  divalglemeuneg  11255  oddpwdclemdvds  11480  oddpwdclemndvds  11481
  Copyright terms: Public domain W3C validator