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Theorem dveeq2 1737
Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveeq2  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Distinct variable group:    x, z

Proof of Theorem dveeq2
StepHypRef Expression
1 ax-i12 1439 . . . . 5  |-  ( A. x  x  =  z  \/  ( A. x  x  =  y  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
2 orcom 680 . . . . . 6  |-  ( ( A. x  x  =  y  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  <->  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y ) )
32orbi2i 712 . . . . 5  |-  ( ( 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
) ) )
41, 3mpbi 143 . . . 4  |-  ( A. x  x  =  z  \/  ( A. x ( z  =  y  ->  A. x  z  =  y )  \/  A. x  x  =  y
) )
5 orass 717 . . . 4  |-  ( ( ( 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 ) ) )
64, 5mpbir 144 . . 3  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  \/ 
A. x  x  =  y )
7 orel2 678 . . 3  |-  ( -. 
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, 7mpi 15 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. x  x  =  z  \/  A. x ( z  =  y  ->  A. x  z  =  y )
) )
9 ax16 1735 . . 3  |-  ( A. x  x  =  z  ->  ( z  =  y  ->  A. x  z  =  y ) )
10 sp 1442 . . 3  |-  ( A. x ( z  =  y  ->  A. x  z  =  y )  ->  ( z  =  y  ->  A. x  z  =  y ) )
119, 10jaoi 669 . 2  |-  ( ( A. x  x  =  z  \/  A. x
( z  =  y  ->  A. x  z  =  y ) )  -> 
( z  =  y  ->  A. x  z  =  y ) )
128, 11syl 14 1  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662   A.wal 1283
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-in2 578  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 1687
This theorem is referenced by:  nd5  1740  ax11v2  1742  dveeq1  1937
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