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Theorem dvelimhw 1876
Description: Proof of dvelimh 2067 without using ax-12 1950 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 3-Mar-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 1629 . . . 4  |-  F/ z  -.  A. x  x  =  y
2 equcom 1692 . . . . . 6  |-  ( z  =  y  <->  y  =  z )
3 nfa1 1806 . . . . . . . 8  |-  F/ x A. x  x  =  y
43nfn 1811 . . . . . . 7  |-  F/ x  -.  A. x  x  =  y
5 dvelimhw.4 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
64, 5nfd 1782 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
72, 6nfxfrd 1580 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x  z  =  y )
8 dvelimhw.1 . . . . . . 7  |-  ( ph  ->  A. x ph )
98nfi 1560 . . . . . 6  |-  F/ x ph
109a1i 11 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
117, 10nfimd 1827 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x
( z  =  y  ->  ph ) )
121, 11nfald 1871 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
13 dvelimhw.2 . . . . 5  |-  ( ps 
->  A. z ps )
14 dvelimhw.3 . . . . 5  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
1513, 14equsalhw 1860 . . . 4  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1615nfbii 1578 . . 3  |-  ( F/ x A. z ( z  =  y  ->  ph )  <->  F/ x ps )
1712, 16sylib 189 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
1817nfrd 1779 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553
This theorem is referenced by:  ax12olem6OLD  2016
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
This theorem depends on definitions:  df-bi 178  df-tru 1328  df-ex 1551  df-nf 1554
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