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Theorem ax10lem2 1890
Description: Lemma for ax10 1897. Change free variable. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax10lem2  |-  ( A. x  x  =  y  ->  A. x  x  =  z )
Distinct variable groups:    x, y    x, z

Proof of Theorem ax10lem2
StepHypRef Expression
1 hbe1 1717 . . . 4  |-  ( E. x  -.  x  =  y  ->  A. x E. x  -.  x  =  y )
2 equequ2 1669 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
32biimprd 214 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  y  ->  x  =  z )
)
43con3rr3 128 . . . . . 6  |-  ( -.  x  =  z  -> 
( z  =  y  ->  -.  x  =  y ) )
5 19.8a 1730 . . . . . 6  |-  ( -.  x  =  y  ->  E. x  -.  x  =  y )
64, 5syl6 29 . . . . 5  |-  ( -.  x  =  z  -> 
( z  =  y  ->  E. x  -.  x  =  y ) )
7 ax-17 1606 . . . . . 6  |-  ( -.  z  =  y  ->  A. x  -.  z  =  y )
8 equequ1 1667 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
98notbid 285 . . . . . . 7  |-  ( x  =  z  ->  ( -.  x  =  y  <->  -.  z  =  y ) )
109biimprd 214 . . . . . 6  |-  ( x  =  z  ->  ( -.  z  =  y  ->  -.  x  =  y ) )
117, 10spimeh 1734 . . . . 5  |-  ( -.  z  =  y  ->  E. x  -.  x  =  y )
126, 11pm2.61d1 151 . . . 4  |-  ( -.  x  =  z  ->  E. x  -.  x  =  y )
131, 12exlimih 1741 . . 3  |-  ( E. x  -.  x  =  z  ->  E. x  -.  x  =  y
)
14 exnal 1564 . . 3  |-  ( E. x  -.  x  =  z  <->  -.  A. x  x  =  z )
15 exnal 1564 . . 3  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
1613, 14, 153imtr3i 256 . 2  |-  ( -. 
A. x  x  =  z  ->  -.  A. x  x  =  y )
1716con4i 122 1  |-  ( A. x  x  =  y  ->  A. x  x  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531
This theorem is referenced by:  ax10lem3  1891
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-ex 1532
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