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Theorem equvin 2027
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equvin  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem equvin
StepHypRef Expression
1 equvini 2013 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 equtr 1689 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
32imp 419 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
43exlimiv 1641 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  x  =  y )
51, 4impbii 181 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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