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Theorem dvelimfALT 1907
Description: Proof of dvelimh 1941 that uses ax-10o 2082 (in the form of ax10o 1894) but not ax-11o 2084, ax-10 2083, or ax-11 1716 (if we replace uses of ax10o 1894 by ax-10o 2082 in the proofs of referenced theorems). See dvelimALT 2076 for a proof (of the distinct variable version dvelim 1962) that doesn't require ax-10 2083. It is not clear whether a proof is possible that uses ax-10 2083 but avoids ax-11 1716, ax-11o 2084, and ax-10o 2082. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dvelimfALT.1  |-  ( ph  ->  A. x ph )
dvelimfALT.2  |-  ( ps 
->  A. z ps )
dvelimfALT.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimfALT  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)

Proof of Theorem dvelimfALT
StepHypRef Expression
1 hba1 1721 . . . . 5  |-  ( A. z ( z  =  y  ->  ph )  ->  A. z A. z ( z  =  y  ->  ph ) )
2 ax10o 1894 . . . . . 6  |-  ( A. z  z  =  x  ->  ( A. z A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
32alequcoms 1889 . . . . 5  |-  ( A. x  x  =  z  ->  ( A. z A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
41, 3syl5 30 . . . 4  |-  ( A. x  x  =  z  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
54a1d 24 . . 3  |-  ( A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z
( z  =  y  ->  ph ) ) ) )
6 hbnae 1898 . . . . . 6  |-  ( -. 
A. x  x  =  z  ->  A. z  -.  A. x  x  =  z )
7 hbnae 1898 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
86, 7hban 1738 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  A. z
( -.  A. x  x  =  z  /\  -.  A. x  x  =  y ) )
9 hbnae 1898 . . . . . . 7  |-  ( -. 
A. x  x  =  z  ->  A. x  -.  A. x  x  =  z )
10 hbnae 1898 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
119, 10hban 1738 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  A. x
( -.  A. x  x  =  z  /\  -.  A. x  x  =  y ) )
12 ax12o 1877 . . . . . . 7  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
1312imp 420 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( z  =  y  ->  A. x  z  =  y )
)
14 dvelimfALT.1 . . . . . . 7  |-  ( ph  ->  A. x ph )
1514a1i 12 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( ph  ->  A. x ph )
)
1611, 13, 15hbimd 1723 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
178, 16hbald 1715 . . . 4  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
1817ex 425 . . 3  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( A. z
( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) ) )
195, 18pm2.61i 158 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
20 dvelimfALT.2 . . 3  |-  ( ps 
->  A. z ps )
21 dvelimfALT.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
2220, 21equsalh 1904 . 2  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
2322albii 1554 . 2  |-  ( A. x A. z ( z  =  y  ->  ph )  <->  A. x ps )
2419, 22, 233imtr3g 262 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528
This theorem is referenced by:  dvelimh  1941  dveeq1ALT  2130  dveel2ALT  2133  a9e2ndVD  27953
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533
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