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Theorem dvelimdc 2545
Description: Deduction form of dvelimc 2546. (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 1626 . . 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 2538 . . . . 5  |-  ( ph  ->  F/ x  w  e.  A )
6 dvelimdc.4 . . . . . 6  |-  ( ph  -> 
F/_ z B )
76nfcrd 2538 . . . . 5  |-  ( ph  ->  F/ z  w  e.  B )
8 dvelimdc.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
9 eleq2 2450 . . . . . 6  |-  ( A  =  B  ->  (
w  e.  A  <->  w  e.  B ) )
108, 9syl6 31 . . . . 5  |-  ( ph  ->  ( z  =  y  ->  ( w  e.  A  <->  w  e.  B
) ) )
112, 3, 5, 7, 10dvelimdf 2117 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x  w  e.  B
) )
1211imp 419 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  w  e.  B )
131, 12nfcd 2520 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x B )
1413ex 424 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2512
This theorem is referenced by:  dvelimc  2546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2382  df-clel 2385  df-nfc 2514
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