Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  a12study11 Unicode version

Theorem a12study11 29138
Description: Experiment to study ax12o 1875. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
a12study11.1  |-  ( -.  z  =  x  -> 
( x  =  y  ->  A. z  x  =  y ) )
Assertion
Ref Expression
a12study11  |-  ( E. z  x  =  y  ->  A. z ( z  =  x  ->  x  =  y ) )
Distinct variable groups:    x, y    x, z

Proof of Theorem a12study11
StepHypRef Expression
1 hba1 1719 . 2  |-  ( A. z ( z  =  x  ->  x  =  y )  ->  A. z A. z ( z  =  x  ->  x  =  y ) )
2 19.8a 1718 . . . . 5  |-  ( ( z  =  x  /\  x  =  y )  ->  E. z ( z  =  x  /\  x  =  y ) )
3 a12study10 29136 . . . . 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 a12study11.1 . . . 4  |-  ( -.  z  =  x  -> 
( x  =  y  ->  A. z  x  =  y ) )
7 ax-1 5 . . . . 5  |-  ( x  =  y  ->  (
z  =  x  ->  x  =  y )
)
87alimi 1546 . . . 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 1729 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 1527   E.wex 1528
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