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Theorem dvelimhw 2079
Description: Proof of dvelimh 2185 without using ax-13 2104 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
Hypotheses
Ref Expression
dvelimhw.1  |-  ( ph  ->  A. x ph )
dvelimhw.2  |-  ( ps 
->  A. z ps )
dvelimhw.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
dvelimhw.4  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Assertion
Ref Expression
dvelimhw  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem dvelimhw
StepHypRef Expression
1 nfv 1769 . . . 4  |-  F/ z  -.  A. x  x  =  y
2 equcom 1870 . . . . . 6  |-  ( z  =  y  <->  y  =  z )
3 nfna1 2005 . . . . . . 7  |-  F/ x  -.  A. x  x  =  y
4 dvelimhw.4 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
53, 4nfd 1976 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
62, 5nfxfrd 1705 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
7 dvelimhw.1 . . . . . . 7  |-  ( ph  ->  A. x ph )
87nfi 1682 . . . . . 6  |-  F/ x ph
98a1i 11 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
106, 9nfimd 2020 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x
( z  =  y  ->  ph ) )
111, 10nfald 2053 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
12 dvelimhw.2 . . . . 5  |-  ( ps 
->  A. z ps )
13 dvelimhw.3 . . . . 5  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
1412, 13equsalhw 2047 . . . 4  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1514nfbii 1703 . . 3  |-  ( F/ x A. z ( z  =  y  ->  ph )  <->  F/ x ps )
1611, 15sylib 201 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
1716nfrd 1973 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
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