MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc11n Structured version   Unicode version

Theorem axc11n 2008
Description: Derive set.mm's original ax-c11n 2198 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 2010 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.) Revised to accomodate a modification of axc11nlem 1875. (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 1712 . . 3  |-  E. z 
z  =  x
2 equcomi 1733 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
3 dveeq1 2003 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
42, 3syl5com 30 . . . . 5  |-  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  x  =  z )
)
5 axc112 1874 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. x  x  =  z )
)
63axc11nlem 1875 . . . . . 6  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
75, 6syl6 33 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
84, 7syl9 71 . . . 4  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
98exlimiv 1689 . . 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 111 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  aecom  2010  axi10  2428  wl-ax11-lem3  28546  wl-ax11-lem8  28551  2sb5ndVD  31959  e2ebindVD  31961  e2ebindALT  31978  2sb5ndALT  31981
  Copyright terms: Public domain W3C validator