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Theorem dvelimALT 2010
Description: Version of dvelim 2017 that doesn't use ax-10 1505. Because it has different distinct variable constraints than dvelim 2017 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimALT.1  |-  ( ph  ->  A. x ph )
dvelimALT.2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimALT  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Distinct variable groups:    ps, z    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y)

Proof of Theorem dvelimALT
StepHypRef Expression
1 nfv 1528 . . . 4  |-  F/ z  -.  A. x  x  =  y
2 ax12or 1508 . . . . . . . . 9  |-  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
3 orcom 728 . . . . . . . . . 10  |-  ( ( A. x  x  =  y  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  <->  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) )
43orbi2i 762 . . . . . . . . 9  |-  ( ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )  <->  ( A. x  x  =  z  \/  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y
) ) )
52, 4mpbi 145 . . . . . . . 8  |-  ( A. x  x  =  z  \/  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y
) )
6 orass 767 . . . . . . . 8  |-  ( ( ( A. x  x  =  z  \/  A. x ( z  =  y  ->  A. x  z  =  y )
)  \/  A. x  x  =  y )  <->  ( A. x  x  =  z  \/  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) ) )
75, 6mpbir 146 . . . . . . 7  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  \/ 
A. x  x  =  y )
8 nfa1 1541 . . . . . . . . . . 11  |-  F/ x A. x  x  =  z
9 ax16ALT 1859 . . . . . . . . . . 11  |-  ( A. x  x  =  z  ->  ( z  =  y  ->  A. x  z  =  y ) )
108, 9nfd 1523 . . . . . . . . . 10  |-  ( A. x  x  =  z  ->  F/ x  z  =  y )
11 dvelimALT.1 . . . . . . . . . . . 12  |-  ( ph  ->  A. x ph )
1211nfi 1462 . . . . . . . . . . 11  |-  F/ x ph
1312a1i 9 . . . . . . . . . 10  |-  ( A. x  x  =  z  ->  F/ x ph )
1410, 13nfimd 1585 . . . . . . . . 9  |-  ( A. x  x  =  z  ->  F/ x ( z  =  y  ->  ph )
)
15 df-nf 1461 . . . . . . . . . 10  |-  ( F/ x  z  =  y  <->  A. x ( z  =  y  ->  A. x  z  =  y )
)
16 id 19 . . . . . . . . . . 11  |-  ( F/ x  z  =  y  ->  F/ x  z  =  y )
1712a1i 9 . . . . . . . . . . 11  |-  ( F/ x  z  =  y  ->  F/ x ph )
1816, 17nfimd 1585 . . . . . . . . . 10  |-  ( F/ x  z  =  y  ->  F/ x ( z  =  y  ->  ph ) )
1915, 18sylbir 135 . . . . . . . . 9  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  F/ x ( z  =  y  ->  ph )
)
2014, 19jaoi 716 . . . . . . . 8  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  ->  F/ x ( z  =  y  ->  ph ) )
2120orim1i 760 . . . . . . 7  |-  ( ( ( A. x  x  =  z  \/  A. x ( z  =  y  ->  A. x  z  =  y )
)  \/  A. x  x  =  y )  ->  ( F/ x ( z  =  y  ->  ph )  \/  A. x  x  =  y )
)
227, 21ax-mp 5 . . . . . 6  |-  ( F/ x ( z  =  y  ->  ph )  \/ 
A. x  x  =  y )
23 orcom 728 . . . . . 6  |-  ( ( F/ x ( z  =  y  ->  ph )  \/  A. x  x  =  y )  <->  ( A. x  x  =  y  \/  F/ x ( z  =  y  ->  ph )
) )
2422, 23mpbi 145 . . . . 5  |-  ( A. x  x  =  y  \/  F/ x ( z  =  y  ->  ph )
)
2524ori 723 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x
( z  =  y  ->  ph ) )
261, 25nfald 1760 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
27 ax-17 1526 . . . . 5  |-  ( ps 
->  A. z ps )
28 dvelimALT.2 . . . . 5  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
2927, 28equsalh 1726 . . . 4  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
3029nfbii 1473 . . 3  |-  ( F/ x A. z ( z  =  y  ->  ph )  <->  F/ x ps )
3126, 30sylib 122 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
3231nfrd 1520 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ wo 708   A.wal 1351   F/wnf 1460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  hbsb4  2012
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