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Theorem equvin 2021
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 1930 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 equtr 1656 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
32imp 420 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
43exlimiv 1670 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  x  =  y )
51, 4impbii 182 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1533
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537
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