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Theorem dvelimdf 2071
Description: Deduction form of dvelimf 2069. This version may be useful if we want to avoid ax-17 1627 and use ax-16 2223 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimdf.1  |-  F/ x ph
dvelimdf.2  |-  F/ z
ph
dvelimdf.3  |-  ( ph  ->  F/ x ps )
dvelimdf.4  |-  ( ph  ->  F/ z ch )
dvelimdf.5  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
dvelimdf  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.1 . . . 4  |-  F/ x ph
2 dvelimdf.3 . . . 4  |-  ( ph  ->  F/ x ps )
31, 2nfim1 1831 . . 3  |-  F/ x
( ph  ->  ps )
4 dvelimdf.2 . . . 4  |-  F/ z
ph
5 dvelimdf.4 . . . 4  |-  ( ph  ->  F/ z ch )
64, 5nfim1 1831 . . 3  |-  F/ z ( ph  ->  ch )
7 dvelimdf.5 . . . . 5  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
87com12 30 . . . 4  |-  ( z  =  y  ->  ( ph  ->  ( ps  <->  ch )
) )
98pm5.74d 240 . . 3  |-  ( z  =  y  ->  (
( ph  ->  ps )  <->  (
ph  ->  ch ) ) )
103, 6, 9dvelimf 2069 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x
( ph  ->  ch )
)
11 pm5.5 328 . . 3  |-  ( ph  ->  ( ( ph  ->  ch )  <->  ch ) )
121, 11nfbidf 1791 . 2  |-  ( ph  ->  ( F/ x (
ph  ->  ch )  <->  F/ x ch ) )
1310, 12syl5ib 212 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550   F/wnf 1554
This theorem is referenced by:  nfsb4t  2129  dvelimdc  2594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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