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Theorem dvelimdf 2022
Description: Deduction form of dvelimf 1937. This version may be useful if we want to avoid ax-17 1603 and use ax-16 2083 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
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.2 . . . . . 6  |-  F/ z
ph
2 dvelimdf.3 . . . . . 6  |-  ( ph  ->  F/ x ps )
31, 2alrimi 1745 . . . . 5  |-  ( ph  ->  A. z F/ x ps )
4 nfsb4t 2020 . . . . 5  |-  ( A. z F/ x ps  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  /  z ] ps ) )
53, 4syl 15 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x [ y  / 
z ] ps )
)
65imp 418 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x [ y  /  z ] ps )
7 dvelimdf.1 . . . . 5  |-  F/ x ph
8 nfnae 1896 . . . . 5  |-  F/ x  -.  A. x  x  =  y
97, 8nfan 1771 . . . 4  |-  F/ x
( ph  /\  -.  A. x  x  =  y
)
10 dvelimdf.4 . . . . . 6  |-  ( ph  ->  F/ z ch )
11 dvelimdf.5 . . . . . 6  |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch )
) )
121, 10, 11sbied 1976 . . . . 5  |-  ( ph  ->  ( [ y  / 
z ] ps  <->  ch )
)
1312adantr 451 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  ( [ y  / 
z ] ps  <->  ch )
)
149, 13nfbidf 1754 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  ( F/ x [
y  /  z ] ps  <->  F/ x ch )
)
156, 14mpbid 201 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ch )
1615ex 423 1  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531   [wsb 1629
This theorem is referenced by:  dvelimdc  2439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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