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Theorem axc11nALT 2154
Description: Alternate proof of axc11n 2153 from axc11nlemALT 2152. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11nALT  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem axc11nALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equcomi 1871 . . . . 5  |-  ( z  =  x  ->  x  =  z )
2 dveeq1 2148 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
31, 2syl5com 31 . . . 4  |-  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  x  =  z )
)
4 axc112 2030 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. x  x  =  z )
)
5 axc11nlemALT 2152 . . . . 5  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
64, 5syl6 34 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
73, 6syl9 73 . . 3  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
8 ax6ev 1817 . . 3  |-  E. z 
z  =  x
97, 8exlimiiv 1787 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. y 
y  =  x  ->  A. y  y  =  x ) )
109pm2.18d 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-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by: (None)
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