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Theorem sbeqalbi 27703
Description: When both  x and  z and  y and  z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
Assertion
Ref Expression
sbeqalbi  |-  ( x  =  y  <->  A. z
( z  =  x  ->  z  =  y ) )
Distinct variable groups:    y, z    x, z

Proof of Theorem sbeqalbi
StepHypRef Expression
1 equtrr 1672 . . 3  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
21alrimiv 1621 . 2  |-  ( x  =  y  ->  A. z
( z  =  x  ->  z  =  y ) )
3 sbeqal1 27700 . 2  |-  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y )
42, 3impbii 180 1  |-  ( x  =  y  <->  A. z
( z  =  x  ->  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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