MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvelimdc Structured version   Unicode version

Theorem dvelimdc 2591
Description: Deduction form of dvelimc 2592. (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 1629 . . 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 2584 . . . . 5  |-  ( ph  ->  F/ x  w  e.  A )
6 dvelimdc.4 . . . . . 6  |-  ( ph  -> 
F/_ z B )
76nfcrd 2584 . . . . 5  |-  ( ph  ->  F/ z  w  e.  B )
8 dvelimdc.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  A  =  B ) )
9 eleq2 2496 . . . . . 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 2066 . . . 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 2566 . 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 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558
This theorem is referenced by:  dvelimc  2592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-cleq 2428  df-clel 2431  df-nfc 2560
  Copyright terms: Public domain W3C validator