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Theorem sbeqal1 26929
Description: If  x  =  y always implies  x  =  z, then 
y  =  z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
Assertion
Ref Expression
sbeqal1  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
Distinct variable group:    x, z

Proof of Theorem sbeqal1
StepHypRef Expression
1 sb2 1889 . 2  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  [ y  /  x ] x  =  z )
2 equsb3 2065 . 2  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
31, 2sylib 190 1  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    = wceq 1619   [wsb 1883
This theorem is referenced by:  sbeqal1i  26930  sbeqalbi  26932
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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