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Theorem sbeqal1 27565
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 2086 . 2  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  [ y  /  x ] x  =  z )
2 equsb3 2177 . 2  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
31, 2sylib 189 1  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   [wsb 1658
This theorem is referenced by:  sbeqal1i  27566  sbeqalbi  27568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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