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Theorem sbeqalbi 27600
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 1653 . . 3  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
21alrimiv 1617 . 2  |-  ( x  =  y  ->  A. z
( z  =  x  ->  z  =  y ) )
3 sbeqal1 27597 . 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 1527    = wceq 1623
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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