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Theorem equvin 2083
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 2079 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 equtr 1694 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
32imp 419 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
43exlimiv 1644 . 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 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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