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Theorem wl-ax11-lem1 30268
Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem1  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  <->  A. y 
y  =  z ) )

Proof of Theorem wl-ax11-lem1
StepHypRef Expression
1 wl-aetr 30226 . 2  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. y 
y  =  z ) )
2 wl-aetr 30226 . . 3  |-  ( A. y  y  =  x  ->  ( A. y  y  =  z  ->  A. x  x  =  z )
)
32aecoms 2056 . 2  |-  ( A. x  x  =  y  ->  ( A. y  y  =  z  ->  A. x  x  =  z )
)
41, 3impbid 191 1  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  <->  A. y 
y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by:  wl-ax11-lem8  30275
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