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Theorem sbeqalbi 36603
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 1846 . . 3  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
21alrimiv 1763 . 2  |-  ( x  =  y  ->  A. z
( z  =  x  ->  z  =  y ) )
3 sbeqal1 36600 . 2  |-  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y )
42, 3impbii 190 1  |-  ( x  =  y  <->  A. z
( z  =  x  ->  z  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by: (None)
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