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Theorem bj-axc11nv 31392
Description: Version of axc11n 2153 with a dv condition, which does not require ax-13 2101. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc11nv  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Distinct variable group:    x, y

Proof of Theorem bj-axc11nv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equcomi 1871 . . . . 5  |-  ( z  =  x  ->  x  =  z )
2 ax-5 1768 . . . . . 6  |-  ( x  =  z  ->  A. y  x  =  z )
32a1i 11 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
41, 3syl5com 31 . . . 4  |-  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  x  =  z )
)
5 axc112 2030 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. x  x  =  z )
)
6 bj-axc11nlemv 31391 . . . . 5  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
75, 6syl6 34 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
84, 7syl9 73 . . 3  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
9 ax6ev 1817 . . 3  |-  E. z 
z  =  x
108, 9exlimiiv 1787 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. y 
y  =  x  ->  A. y  y  =  x ) )
1110pm2.18d 116 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  bj-aecomsv  31393  bj-naecomsv  31394
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