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Theorem a12study11n 29761
Description: Experiment to study ax12o 1887. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
a12study11n.1  |-  ( -.  z  =  x  -> 
( -.  x  =  y  ->  A. z  -.  x  =  y
) )
Assertion
Ref Expression
a12study11n  |-  ( E. z  -.  x  =  y  ->  A. z
( z  =  x  ->  -.  x  =  y ) )
Distinct variable groups:    x, y    x, z

Proof of Theorem a12study11n
StepHypRef Expression
1 hba1 1731 . 2  |-  ( A. z ( z  =  x  ->  -.  x  =  y )  ->  A. z A. z ( z  =  x  ->  -.  x  =  y
) )
2 19.8a 1730 . . . . 5  |-  ( ( z  =  x  /\  -.  x  =  y
)  ->  E. z
( z  =  x  /\  -.  x  =  y ) )
3 a12study10n 29759 . . . . 5  |-  ( E. z ( z  =  x  /\  -.  x  =  y )  ->  A. z ( z  =  x  ->  -.  x  =  y ) )
42, 3syl 15 . . . 4  |-  ( ( z  =  x  /\  -.  x  =  y
)  ->  A. z
( z  =  x  ->  -.  x  =  y ) )
54ex 423 . . 3  |-  ( z  =  x  ->  ( -.  x  =  y  ->  A. z ( z  =  x  ->  -.  x  =  y )
) )
6 a12study11n.1 . . . 4  |-  ( -.  z  =  x  -> 
( -.  x  =  y  ->  A. z  -.  x  =  y
) )
7 ax-1 5 . . . . 5  |-  ( -.  x  =  y  -> 
( z  =  x  ->  -.  x  =  y ) )
87alimi 1549 . . . 4  |-  ( A. z  -.  x  =  y  ->  A. z ( z  =  x  ->  -.  x  =  y )
)
96, 8syl6 29 . . 3  |-  ( -.  z  =  x  -> 
( -.  x  =  y  ->  A. z
( z  =  x  ->  -.  x  =  y ) ) )
105, 9pm2.61i 156 . 2  |-  ( -.  x  =  y  ->  A. z ( z  =  x  ->  -.  x  =  y ) )
111, 10exlimih 1741 1  |-  ( E. z  -.  x  =  y  ->  A. z
( z  =  x  ->  -.  x  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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