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

Theorem dvelimdf 1969
Description: Deduction form of dvelimf 1968. This version may be useful if we want to avoid ax-17 1491 and use ax-16 1770 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimdf.1  |-  F/ x ph
dvelimdf.2  |-  F/ z
ph
dvelimdf.3  |-  ( ph  ->  F/ x ps )
dvelimdf.4  |-  ( ph  ->  F/ z ch )
dvelimdf.5  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
dvelimdf  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.2 . . . 4  |-  F/ z
ph
2 dvelimdf.3 . . . 4  |-  ( ph  ->  F/ x ps )
31, 2alrimi 1487 . . 3  |-  ( ph  ->  A. z F/ x ps )
4 nfsb4t 1967 . . 3  |-  ( A. z F/ x ps  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  /  z ] ps ) )
53, 4syl 14 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  / 
z ] ps )
)
6 dvelimdf.1 . . 3  |-  F/ x ph
7 dvelimdf.4 . . . 4  |-  ( ph  ->  F/ z ch )
8 dvelimdf.5 . . . 4  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
91, 7, 8sbied 1746 . . 3  |-  ( ph  ->  ( [ y  / 
z ] ps  <->  ch )
)
106, 9nfbidf 1504 . 2  |-  ( ph  ->  ( F/ x [
y  /  z ] ps  <->  F/ x ch )
)
115, 10sylibd 148 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104   A.wal 1314   F/wnf 1421   [wsb 1720
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721
This theorem is referenced by:  dvelimdc  2278
  Copyright terms: Public domain W3C validator