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Theorem dvelimdc 2301
Description: Deduction form of dvelimc 2302. (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 1508 . . 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 2295 . . . . 5  |-  ( ph  ->  F/ x  w  e.  A )
6 dvelimdc.4 . . . . . 6  |-  ( ph  -> 
F/_ z B )
76nfcrd 2295 . . . . 5  |-  ( ph  ->  F/ z  w  e.  B )
8 dvelimdc.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
9 eleq2 2203 . . . . . 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 1991 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x  w  e.  B
) )
1211imp 123 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  w  e.  B )
131, 12nfcd 2276 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x B )
1413ex 114 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331   F/wnf 1436    e. wcel 1480   F/_wnfc 2268
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270
This theorem is referenced by:  dvelimc  2302
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