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Theorem dvelimdc 2452
Description: Deduction form of dvelimc 2453. (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 1609 . . 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 2445 . . . . 5  |-  ( ph  ->  F/ x  w  e.  A )
6 dvelimdc.4 . . . . . 6  |-  ( ph  -> 
F/_ z B )
76nfcrd 2445 . . . . 5  |-  ( ph  ->  F/ z  w  e.  B )
8 dvelimdc.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
9 eleq2 2357 . . . . . 6  |-  ( A  =  B  ->  (
w  e.  A  <->  w  e.  B ) )
108, 9syl6 29 . . . . 5  |-  ( ph  ->  ( z  =  y  ->  ( w  e.  A  <->  w  e.  B
) ) )
112, 3, 5, 7, 10dvelimdf 2035 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x  w  e.  B
) )
1211imp 418 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  w  e.  B )
131, 12nfcd 2427 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x B )
1413ex 423 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419
This theorem is referenced by:  dvelimc  2453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421
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