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Theorem equvelv 29738
Description: Similar to equveli 1941 without using ax12o 1887. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equvelv  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  x  =  y
)
Distinct variable groups:    x, z    y, z

Proof of Theorem equvelv
StepHypRef Expression
1 albiim 1601 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) ) )
2 equveliv 29737 . . . 4  |-  ( A. z ( z  =  x  ->  z  =  y )  <->  x  =  y )
3 equveliv 29737 . . . . 5  |-  ( A. z ( z  =  y  ->  z  =  x )  <->  y  =  x )
4 equcomi 1664 . . . . . 6  |-  ( y  =  x  ->  x  =  y )
5 equcomi 1664 . . . . . 6  |-  ( x  =  y  ->  y  =  x )
64, 5impbii 180 . . . . 5  |-  ( y  =  x  <->  x  =  y )
73, 6bitri 240 . . . 4  |-  ( A. z ( z  =  y  ->  z  =  x )  <->  x  =  y )
82, 7anbi12i 678 . . 3  |-  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  <->  ( x  =  y  /\  x  =  y ) )
9 anidm 625 . . 3  |-  ( ( x  =  y  /\  x  =  y )  <->  x  =  y )
108, 9bitri 240 . 2  |-  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  <->  x  =  y )
111, 10bitri 240 1  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  x  =  y
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530
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-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360
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