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Theorem dvelimor 1910
Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula  ph (containing  z) and a distinct variable constraint between 
x and  z. The theorem makes it possible to replace the distinct variable constraint with the disjunct  A. x x  =  y ( ps is just a version of  ph with  y substituted for  z). (Contributed by Jim Kingdon, 11-May-2018.)
Hypotheses
Ref Expression
dvelimor.1  |-  F/ x ph
dvelimor.2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimor  |-  ( A. x  x  =  y  \/  F/ x ps )
Distinct variable groups:    ps, z    x, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y)

Proof of Theorem dvelimor
StepHypRef Expression
1 ax-bndl 1415 . . . . . 6  |-  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
) )
2 orcom 657 . . . . . . 7  |-  ( ( A. x  x  =  y  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
)  <->  ( A. z A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) )
32orbi2i 689 . . . . . 6  |-  ( ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. z A. x
( z  =  y  ->  A. x  z  =  y ) ) )  <-> 
( A. x  x  =  z  \/  ( A. z A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y
) ) )
41, 3mpbi 137 . . . . 5  |-  ( A. x  x  =  z  \/  ( A. z A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) )
5 orass 694 . . . . 5  |-  ( ( ( A. x  x  =  z  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
)  \/  A. x  x  =  y )  <->  ( A. x  x  =  z  \/  ( A. z A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) ) )
64, 5mpbir 138 . . . 4  |-  ( ( A. x  x  =  z  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
)  \/  A. x  x  =  y )
7 nfae 1623 . . . . . . 7  |-  F/ z A. x  x  =  z
8 a16nf 1762 . . . . . . 7  |-  ( A. x  x  =  z  ->  F/ x ( z  =  y  ->  ph )
)
97, 8alrimi 1431 . . . . . 6  |-  ( A. x  x  =  z  ->  A. z F/ x
( z  =  y  ->  ph ) )
10 df-nf 1366 . . . . . . . 8  |-  ( F/ x  z  =  y  <->  A. x ( z  =  y  ->  A. x  z  =  y )
)
11 id 19 . . . . . . . . 9  |-  ( F/ x  z  =  y  ->  F/ x  z  =  y )
12 dvelimor.1 . . . . . . . . . 10  |-  F/ x ph
1312a1i 9 . . . . . . . . 9  |-  ( F/ x  z  =  y  ->  F/ x ph )
1411, 13nfimd 1493 . . . . . . . 8  |-  ( F/ x  z  =  y  ->  F/ x ( z  =  y  ->  ph ) )
1510, 14sylbir 129 . . . . . . 7  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  F/ x ( z  =  y  ->  ph )
)
1615alimi 1360 . . . . . 6  |-  ( A. z A. x ( z  =  y  ->  A. x  z  =  y )  ->  A. z F/ x
( z  =  y  ->  ph ) )
179, 16jaoi 646 . . . . 5  |-  ( ( A. x  x  =  z  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
)  ->  A. z F/ x ( z  =  y  ->  ph ) )
1817orim1i 687 . . . 4  |-  ( ( ( A. x  x  =  z  \/  A. z A. x ( z  =  y  ->  A. x  z  =  y )
)  \/  A. x  x  =  y )  ->  ( A. z F/ x ( z  =  y  ->  ph )  \/ 
A. x  x  =  y ) )
196, 18ax-mp 7 . . 3  |-  ( A. z F/ x ( z  =  y  ->  ph )  \/  A. x  x  =  y )
20 orcom 657 . . 3  |-  ( ( A. z F/ x
( z  =  y  ->  ph )  \/  A. x  x  =  y
)  <->  ( A. x  x  =  y  \/  A. z F/ x ( z  =  y  ->  ph ) ) )
2119, 20mpbi 137 . 2  |-  ( A. x  x  =  y  \/  A. z F/ x
( z  =  y  ->  ph ) )
22 nfalt 1486 . . . 4  |-  ( A. z F/ x ( z  =  y  ->  ph )  ->  F/ x A. z
( z  =  y  ->  ph ) )
23 ax-17 1435 . . . . . 6  |-  ( ps 
->  A. z ps )
24 dvelimor.2 . . . . . 6  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
2523, 24equsalh 1630 . . . . 5  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
2625nfbii 1378 . . . 4  |-  ( F/ x A. z ( z  =  y  ->  ph )  <->  F/ x ps )
2722, 26sylib 131 . . 3  |-  ( A. z F/ x ( z  =  y  ->  ph )  ->  F/ x ps )
2827orim2i 688 . 2  |-  ( ( A. x  x  =  y  \/  A. z F/ x ( z  =  y  ->  ph ) )  ->  ( A. x  x  =  y  \/  F/ x ps ) )
2921, 28ax-mp 7 1  |-  ( A. x  x  =  y  \/  F/ x ps )
Colors of variables: wff set class
Syntax hints:    -> 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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  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:  nfsb4or  1915  rgen2a  2392
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