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Theorem wl-aetr 31827
Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-aetr  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. y 
y  =  z ) )

Proof of Theorem wl-aetr
StepHypRef Expression
1 ax-7 1843 . . 3  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
21al2imi 1681 . 2  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. x  y  =  z )
)
3 axc11 2113 . 2  |-  ( A. x  x  =  y  ->  ( A. x  y  =  z  ->  A. y 
y  =  z ) )
42, 3syld 45 1  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. y 
y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
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 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  wl-ax11-lem1  31879  wl-ax11-lem3  31881
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