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Theorem a12study5rev 29193
Description: Experiment to study ax12o 1947. The hypothesis is a conjectured ax12o 1947 replacement. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
a12study5rev.1  |-  ( A. y  -.  z  =  x  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) )
Assertion
Ref Expression
a12study5rev  |-  ( -. 
A. z  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y )
)
Distinct variable groups:    x, y    y, z

Proof of Theorem a12study5rev
StepHypRef Expression
1 exnal 1579 . 2  |-  ( E. z  -.  z  =  x  <->  -.  A. z 
z  =  x )
2 19.8a 1752 . . 3  |-  ( x  =  y  ->  E. z  x  =  y )
3 hbe1 1736 . . . . 5  |-  ( E. z  x  =  y  ->  A. z E. z  x  =  y )
4 hba1 1792 . . . . 5  |-  ( A. z  x  =  y  ->  A. z A. z  x  =  y )
53, 4hbim 1824 . . . 4  |-  ( ( E. z  x  =  y  ->  A. z  x  =  y )  ->  A. z ( E. z  x  =  y  ->  A. z  x  =  y ) )
6 ax-17 1621 . . . . 5  |-  ( -.  z  =  x  ->  A. y  -.  z  =  x )
7 df-ex 1547 . . . . . 6  |-  ( E. z  x  =  y  <->  -.  A. z  -.  x  =  y )
8 a12study5rev.1 . . . . . 6  |-  ( A. y  -.  z  =  x  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) )
97, 8syl5bi 208 . . . . 5  |-  ( A. y  -.  z  =  x  ->  ( E. z  x  =  y  ->  A. z  x  =  y ) )
106, 9syl 15 . . . 4  |-  ( -.  z  =  x  -> 
( E. z  x  =  y  ->  A. z  x  =  y )
)
115, 10exlimih 1810 . . 3  |-  ( E. z  -.  z  =  x  ->  ( E. z  x  =  y  ->  A. z  x  =  y ) )
122, 11syl5 28 . 2  |-  ( E. z  -.  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y )
)
131, 12sylbir 204 1  |-  ( -. 
A. z  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1545   E.wex 1546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-11 1751
This theorem depends on definitions:  df-bi 177  df-ex 1547  df-nf 1550
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