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

Theorem dvelimdc 2248
Description: Deduction form of dvelimc 2249. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
dvelimdc.1  |-  F/ x ph
dvelimdc.2  |-  F/ z
ph
dvelimdc.3  |-  ( ph  -> 
F/_ x A )
dvelimdc.4  |-  ( ph  -> 
F/_ z B )
dvelimdc.5  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
Assertion
Ref Expression
dvelimdc  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x B ) )

Proof of Theorem dvelimdc
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . . 3  |-  F/ w
( ph  /\  -.  A. x  x  =  y
)
2 dvelimdc.1 . . . . 5  |-  F/ x ph
3 dvelimdc.2 . . . . 5  |-  F/ z
ph
4 dvelimdc.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
54nfcrd 2242 . . . . 5  |-  ( ph  ->  F/ x  w  e.  A )
6 dvelimdc.4 . . . . . 6  |-  ( ph  -> 
F/_ z B )
76nfcrd 2242 . . . . 5  |-  ( ph  ->  F/ z  w  e.  B )
8 dvelimdc.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
9 eleq2 2151 . . . . . 6  |-  ( A  =  B  ->  (
w  e.  A  <->  w  e.  B ) )
108, 9syl6 33 . . . . 5  |-  ( ph  ->  ( z  =  y  ->  ( w  e.  A  <->  w  e.  B
) ) )
112, 3, 5, 7, 10dvelimdf 1940 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x  w  e.  B
) )
1211imp 122 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  w  e.  B )
131, 12nfcd 2223 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x B )
1413ex 113 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394    e. wcel 1438   F/_wnfc 2215
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217
This theorem is referenced by:  dvelimc  2249
  Copyright terms: Public domain W3C validator