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Theorem sbeqal1 27597
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 1963 . 2  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  [ y  /  x ] x  =  z )
2 equsb3 2041 . 2  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
31, 2sylib 188 1  |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623   [wsb 1629
This theorem is referenced by:  sbeqal1i  27598  sbeqalbi  27600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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