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Theorem sbeqalbi 27591
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 1696 . . 3  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
21alrimiv 1642 . 2  |-  ( x  =  y  ->  A. z
( z  =  x  ->  z  =  y ) )
3 sbeqal1 27588 . 2  |-  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y )
42, 3impbii 182 1  |-  ( x  =  y  <->  A. z
( z  =  x  ->  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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