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Theorem a12study10 29136
Description: Experiment to study ax12o 1875. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12study10  |-  ( E. z ( z  =  x  /\  x  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) )
Distinct variable groups:    x, y    x, z

Proof of Theorem a12study10
StepHypRef Expression
1 ax9v 1636 . . 3  |-  -.  A. z  -.  z  =  x
2 df-ex 1529 . . 3  |-  ( E. z  z  =  x  <->  -.  A. z  -.  z  =  x )
31, 2mpbir 200 . 2  |-  E. z 
z  =  x
4 hbe1 1705 . . . 4  |-  ( E. z ( z  =  x  /\  x  =  y )  ->  A. z E. z ( z  =  x  /\  x  =  y ) )
5 hba1 1719 . . . 4  |-  ( A. z ( z  =  x  ->  x  =  y )  ->  A. z A. z ( z  =  x  ->  x  =  y ) )
64, 5hbim 1725 . . 3  |-  ( ( E. z ( z  =  x  /\  x  =  y )  ->  A. z ( z  =  x  ->  x  =  y ) )  ->  A. z ( E. z
( z  =  x  /\  x  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) ) )
7 ax-17 1603 . . . . . . . 8  |-  ( -.  z  =  y  ->  A. x  -.  z  =  y )
8 ax-11 1715 . . . . . . . 8  |-  ( z  =  x  ->  ( A. x  -.  z  =  y  ->  A. z
( z  =  x  ->  -.  z  =  y ) ) )
97, 8syl5 28 . . . . . . 7  |-  ( z  =  x  ->  ( -.  z  =  y  ->  A. z ( z  =  x  ->  -.  z  =  y )
) )
10 equequ1 1648 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
1110notbid 285 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( -.  z  =  y  <->  -.  x  =  y ) )
1211pm5.74i 236 . . . . . . . . . 10  |-  ( ( z  =  x  ->  -.  z  =  y
)  <->  ( z  =  x  ->  -.  x  =  y ) )
13 imnan 411 . . . . . . . . . 10  |-  ( ( z  =  x  ->  -.  x  =  y
)  <->  -.  ( z  =  x  /\  x  =  y ) )
1412, 13bitri 240 . . . . . . . . 9  |-  ( ( z  =  x  ->  -.  z  =  y
)  <->  -.  ( z  =  x  /\  x  =  y ) )
1514albii 1553 . . . . . . . 8  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  <->  A. z  -.  ( z  =  x  /\  x  =  y ) )
16 alnex 1530 . . . . . . . 8  |-  ( A. z  -.  ( z  =  x  /\  x  =  y )  <->  -.  E. z
( z  =  x  /\  x  =  y ) )
1715, 16bitri 240 . . . . . . 7  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  <->  -.  E. z
( z  =  x  /\  x  =  y ) )
189, 17syl6ib 217 . . . . . 6  |-  ( z  =  x  ->  ( -.  z  =  y  ->  -.  E. z ( z  =  x  /\  x  =  y )
) )
1918con4d 97 . . . . 5  |-  ( z  =  x  ->  ( E. z ( z  =  x  /\  x  =  y )  ->  z  =  y ) )
20 ax-17 1603 . . . . 5  |-  ( z  =  y  ->  A. x  z  =  y )
2119, 20syl6 29 . . . 4  |-  ( z  =  x  ->  ( E. z ( z  =  x  /\  x  =  y )  ->  A. x  z  =  y )
)
22 ax-11 1715 . . . . 5  |-  ( z  =  x  ->  ( A. x  z  =  y  ->  A. z ( z  =  x  ->  z  =  y ) ) )
23 ax-8 1643 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
2423a2i 12 . . . . . 6  |-  ( ( z  =  x  -> 
z  =  y )  ->  ( z  =  x  ->  x  =  y ) )
2524alimi 1546 . . . . 5  |-  ( A. z ( z  =  x  ->  z  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) )
2622, 25syl6 29 . . . 4  |-  ( z  =  x  ->  ( A. x  z  =  y  ->  A. z ( z  =  x  ->  x  =  y ) ) )
2721, 26syld 40 . . 3  |-  ( z  =  x  ->  ( E. z ( z  =  x  /\  x  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) ) )
286, 27exlimih 1729 . 2  |-  ( E. z  z  =  x  ->  ( E. z
( z  =  x  /\  x  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) ) )
293, 28ax-mp 8 1  |-  ( E. z ( z  =  x  /\  x  =  y )  ->  A. z
( z  =  x  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  a12study11  29138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator