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Theorem dveeq2or 1739
Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1738 but connecting  A. x x  =  y by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
dveeq2or  |-  ( A. x  x  =  y  \/  F/ x  z  =  y )
Distinct variable group:    x, z

Proof of Theorem dveeq2or
StepHypRef Expression
1 ax-i12 1439 . . . . . 6  |-  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
2 orass 717 . . . . . 6  |-  ( ( ( A. x  x  =  z  \/  A. x  x  =  y
)  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  <->  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) ) )
31, 2mpbir 144 . . . . 5  |-  ( ( A. x  x  =  z  \/  A. x  x  =  y )  \/  A. x ( z  =  y  ->  A. x  z  =  y )
)
4 pm1.4 679 . . . . . 6  |-  ( ( A. x  x  =  z  \/  A. x  x  =  y )  ->  ( A. x  x  =  y  \/  A. x  x  =  z
) )
54orim1i 710 . . . . 5  |-  ( ( ( A. x  x  =  z  \/  A. x  x  =  y
)  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  -> 
( ( A. x  x  =  y  \/  A. x  x  =  z )  \/  A. x
( z  =  y  ->  A. x  z  =  y ) ) )
63, 5ax-mp 7 . . . 4  |-  ( ( A. x  x  =  y  \/  A. x  x  =  z )  \/  A. x ( z  =  y  ->  A. x  z  =  y )
)
7 orass 717 . . . 4  |-  ( ( ( A. x  x  =  y  \/  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 )
) ) )
86, 7mpbi 143 . . 3  |-  ( A. x  x  =  y  \/  ( A. x  x  =  z  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
9 ax16 1736 . . . . . 6  |-  ( A. x  x  =  z  ->  ( z  =  y  ->  A. x  z  =  y ) )
109a5i 1476 . . . . 5  |-  ( A. x  x  =  z  ->  A. x ( z  =  y  ->  A. x  z  =  y )
)
11 id 19 . . . . 5  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  A. x ( z  =  y  ->  A. x  z  =  y )
)
1210, 11jaoi 669 . . . 4  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  ->  A. x ( z  =  y  ->  A. x  z  =  y )
)
1312orim2i 711 . . 3  |-  ( ( A. x  x  =  y  \/  ( A. x  x  =  z  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )  ->  ( A. x  x  =  y  \/  A. x
( z  =  y  ->  A. x  z  =  y ) ) )
148, 13ax-mp 7 . 2  |-  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
)
15 df-nf 1391 . . . 4  |-  ( F/ x  z  =  y  <->  A. x ( z  =  y  ->  A. x  z  =  y )
)
1615biimpri 131 . . 3  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  F/ x  z  =  y )
1716orim2i 711 . 2  |-  ( ( A. x  x  =  y  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  -> 
( A. x  x  =  y  \/  F/ x  z  =  y
) )
1814, 17ax-mp 7 1  |-  ( A. x  x  =  y  \/  F/ x  z  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662   A.wal 1283   F/wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  equs5or  1753  sbal1yz  1920  copsexg  4027
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