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Theorem dvelimALT 1902
Description: Version of dvelim 1909 that doesn't use ax-10 1412. Because it has different distinct variable constraints than dvelim 1909 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 1437 . . . 4  |-  F/ z  -.  A. x  x  =  y
2 ax-i12 1414 . . . . . . . . 9  |-  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
3 orcom 657 . . . . . . . . . 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 689 . . . . . . . . 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 137 . . . . . . . 8  |-  ( A. x  x  =  z  \/  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y
) )
6 orass 694 . . . . . . . 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 138 . . . . . . 7  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  \/ 
A. x  x  =  y )
8 nfa1 1450 . . . . . . . . . . 11  |-  F/ x A. x  x  =  z
9 ax16ALT 1755 . . . . . . . . . . 11  |-  ( A. x  x  =  z  ->  ( z  =  y  ->  A. x  z  =  y ) )
108, 9nfd 1432 . . . . . . . . . 10  |-  ( A. x  x  =  z  ->  F/ x  z  =  y )
11 dvelimALT.1 . . . . . . . . . . . 12  |-  ( ph  ->  A. x ph )
1211nfi 1367 . . . . . . . . . . 11  |-  F/ x ph
1312a1i 9 . . . . . . . . . 10  |-  ( A. x  x  =  z  ->  F/ x ph )
1410, 13nfimd 1493 . . . . . . . . 9  |-  ( A. x  x  =  z  ->  F/ x ( z  =  y  ->  ph )
)
15 df-nf 1366 . . . . . . . . . 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 1493 . . . . . . . . . 10  |-  ( F/ x  z  =  y  ->  F/ x ( z  =  y  ->  ph ) )
1915, 18sylbir 129 . . . . . . . . 9  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  F/ x ( z  =  y  ->  ph )
)
2014, 19jaoi 646 . . . . . . . 8  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  ->  F/ x ( z  =  y  ->  ph ) )
2120orim1i 687 . . . . . . 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 7 . . . . . 6  |-  ( F/ x ( z  =  y  ->  ph )  \/ 
A. x  x  =  y )
23 orcom 657 . . . . . 6  |-  ( ( F/ x ( z  =  y  ->  ph )  \/  A. x  x  =  y )  <->  ( A. x  x  =  y  \/  F/ x ( z  =  y  ->  ph )
) )
2422, 23mpbi 137 . . . . 5  |-  ( A. x  x  =  y  \/  F/ x ( z  =  y  ->  ph )
)
2524ori 652 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x
( z  =  y  ->  ph ) )
261, 25nfald 1659 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
27 ax-17 1435 . . . . 5  |-  ( ps 
->  A. z ps )
28 dvelimALT.2 . . . . 5  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
2927, 28equsalh 1630 . . . 4  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
3029nfbii 1378 . . 3  |-  ( F/ x A. z ( z  =  y  ->  ph )  <->  F/ x ps )
3126, 30sylib 131 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
3231nfrd 1429 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 102    \/ wo 639   A.wal 1257   F/wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  hbsb4  1904
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