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Theorem dvelimfALT 1853
Description: Proof of dvelimh 1974 that uses ax-10o 1835 (in the form of ax10o 1834) but not ax-11o 1940, ax-10 1678, or ax-11 1624 (if we replace uses of ax10o 1834 by ax-10o 1835 in the proofs of referenced theorems). See dvelimALT 2094 for a proof (of the distinct variable version dvelim 2092) that doesn't require ax-10 1678. It is not clear whether a proof is possible that uses ax-10 1678 but avoids ax-11 1624, ax-11o 1940, and ax-10o 1835. (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 1718 . . . . 5  |-  ( A. z ( z  =  y  ->  ph )  ->  A. z A. z ( z  =  y  ->  ph ) )
2 ax10o 1834 . . . . . 6  |-  ( A. z  z  =  x  ->  ( A. z A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
32alequcoms 1681 . . . . 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 1844 . . . . . 6  |-  ( -. 
A. x  x  =  z  ->  A. z  -.  A. x  x  =  z )
7 hbnae 1844 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
86, 7hban 1724 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  A. z
( -.  A. x  x  =  z  /\  -.  A. x  x  =  y ) )
9 hbnae 1844 . . . . . . 7  |-  ( -. 
A. x  x  =  z  ->  A. x  -.  A. x  x  =  z )
10 hbnae 1844 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
119, 10hban 1724 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  A. x
( -.  A. x  x  =  z  /\  -.  A. x  x  =  y ) )
12 ax-12o 1664 . . . . . . 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 1809 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
178, 16hbald 1614 . . . 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 1851 . 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 1532    = wceq 1619
This theorem is referenced by:  dveeq2  1928  dveeq2ALT  1930  dvelimh  1974  dveeq1ALT  2097  ax15  2101  dveel2ALT  2104  a9e2ndVD  27374
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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