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Theorem axc11n 2110
Description: Derive set.mm's original ax-c11n 32372 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Use aecom 2112 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Adapt to a modification of axc11nlem 1998. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
axc11n  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem axc11n
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1800 . . 3  |-  E. z 
z  =  x
2 equcomi 1847 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
3 dveeq1 2105 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
42, 3syl5com 31 . . . . 5  |-  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  x  =  z )
)
5 axc112 1997 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. x  x  =  z )
)
63axc11nlem 1998 . . . . . 6  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
75, 6syl6 34 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
84, 7syl9 73 . . . 4  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
98exlimiv 1770 . . 3  |-  ( E. z  z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y 
y  =  x ) ) )
101, 9ax-mp 5 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. y 
y  =  x  ->  A. y  y  =  x ) )
1110pm2.18d 114 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2058
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  aecom  2112  axi10  2400  wl-ax11-lem3  31824  wl-ax11-lem8  31829  2sb5ndVD  37223  e2ebindVD  37225  e2ebindALT  37242  2sb5ndALT  37245
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